{"id":4238,"date":"2026-04-05T15:28:44","date_gmt":"2026-04-05T06:28:44","guid":{"rendered":"http:\/\/batmask.net\/?page_id=4238"},"modified":"2026-04-24T18:42:17","modified_gmt":"2026-04-24T09:42:17","slug":"differential-equation","status":"publish","type":"page","link":"http:\/\/batmask.net\/index.php\/differential-equation\/","title":{"rendered":"Differential Equation"},"content":{"rendered":"\n<ul class=\"wp-block-list\">\n<li><a href=\"#directlyintegrable-de\" data-type=\"internal\" data-id=\"#directlyintegrable-de\">Directly Integrable D.E.<\/a><\/li>\n\n\n\n<li><a href=\"#separable-de\" data-type=\"internal\" data-id=\"#separable-de\">Separable D.E.<\/a><\/li>\n\n\n\n<li><a href=\"#1st_order-integ-factor\" data-type=\"internal\" data-id=\"#1st_order-integ-factor\">Linear 1st order D.E. : Integrating factor<\/a><\/li>\n\n\n\n<li><a href=\"#homogeneous-1st-order-de\" data-type=\"internal\" data-id=\"#homogeneous-1st-order-de\">Homogeneous 1st order D.E.<\/a><\/li>\n\n\n\n<li><a href=\"#bernoui-de\" data-type=\"internal\" data-id=\"#bernoui-de\">Bernoui D.E.<\/a><\/li>\n\n\n\n<li><a href=\"#exact-de\" data-type=\"internal\" data-id=\"#exact-de\">Exact D.E.<\/a><\/li>\n\n\n\n<li><a href=\"#linear-2nd-order-de\" data-type=\"internal\" data-id=\"#linear-2nd-order-de\">Linear 2nd order D.E.<\/a><\/li>\n\n\n\n<li><a href=\"#reduction-of-order\" data-type=\"internal\" data-id=\"#reduction-of-order\">2nd order D.E. : Reduction of order<\/a><\/li>\n\n\n\n<li><a href=\"#constant-coefficients\" data-type=\"internal\" data-id=\"#constant-coefficients\">2nd order D.E. : Constant Coefficients<\/a><\/li>\n\n\n\n<li><a href=\"#undetermined-coefficients\" data-type=\"internal\" data-id=\"#undetermined-coefficients\">2nd order D.E. : Undetermined Coefficients<\/a><\/li>\n\n\n\n<li><a href=\"#wronskian\" data-type=\"internal\" data-id=\"#wronskian\">Wronskian For a Group of Functions<\/a><\/li>\n\n\n\n<li><a href=\"#variation-of-parameters\" data-type=\"internal\" data-id=\"#variation-of-parameters\">2nd order D.E. : Variation of Parameters<\/a><\/li>\n\n\n\n<li><a href=\"#cauchy-euler-eq\" data-type=\"internal\" data-id=\"#cauchy-euler-eq\">Cauchy-Euler Eq.<\/a><\/li>\n\n\n\n<li><a href=\"#laplace-transform\" data-type=\"internal\" data-id=\"#laplace-transform\">Laplace Transform<\/a><\/li>\n\n\n\n<li><a href=\"#Linearity-of-Laplace-Transforms\" data-type=\"internal\" data-id=\"#Linearity-of-Laplace-Transforms\">Linearity of Laplace Transforms<\/a><\/li>\n\n\n\n<li><a href=\"#Laplace-Transforms-of-Derivatives\" data-type=\"internal\" data-id=\"#Laplace-Transforms-of-Derivatives\">Laplace Transforms of Derivatives<\/a><\/li>\n\n\n\n<li><a href=\"#Laplace-Transforms-to-Solve-D.E.\" data-type=\"internal\" data-id=\"#Laplace-Transforms-to-Solve-D.E.\">Laplace Transforms to Solve D.E.<\/a><\/li>\n\n\n\n<li><a href=\"#Unit-Step-Function\" data-type=\"internal\" data-id=\"#Unit-Step-Function\">Unit Step Function<\/a><\/li>\n\n\n\n<li><a href=\"#Laplace-Transforms-Involving-the-Unit-Step-Function\" data-type=\"internal\" data-id=\"#Laplace-Transforms-Involving-the-Unit-Step-Function\">Laplace Transforms Involving the Unit Step Function<\/a><\/li>\n\n\n\n<li><a href=\"#Impulse-and-Dirac-Delta-Function\" data-type=\"internal\" data-id=\"#Impulse-and-Dirac-Delta-Function\">Impulse and Dirac Delta Function<\/a><\/li>\n\n\n\n<li><a href=\"#How-to-Shift-the-Index-for-Power-Series\" data-type=\"internal\" data-id=\"#How-to-Shift-the-Index-for-Power-Series\">How to shift the Index for Power Series<\/a><\/li>\n\n\n\n<li><a href=\"#Solving-D.E.-with-Power-Series\" data-type=\"internal\" data-id=\"#Solving-D.E.-with-Power-Series\">Solving D.E. with Power Series<\/a><\/li>\n\n\n\n<li><a href=\"#Solving-D.E-with-eigenvalue-and-eigenvector\" data-type=\"internal\" data-id=\"#Solving-D.E-with-eigenvalue-and-eigenvector\">Solving D.E. Linear system with eigenvalue and eigenvector<\/a><\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"directlyintegrable-de\">Directly Integrable D.E.<\/h2>\n\n\n\n<p>\uadf8\ub0e5 \uc801\ubd84\ud558\uba74 \ub418\ub294 \ud615\ud0dc.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{matrix}\n\\begin{align*}\ny^\\prime = f(x)  \\\\\ny^{\\prime\\prime} = f(x) \\\\\n\\end{align*} &amp; \\quad\n\\begin{align*}\nex)\\quad  \\frac{dy}{dx} = 6x^2 + 4\n\\end{align*}\n\\end {matrix}<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"separable-de\">Separable D.E.<\/h2>\n\n\n\n<p>\uc67c\ucabd\uc740 y\uc5d0 \ub300\ud574\uc11c\ub9cc, \uc624\ub978\ucabd x\uc5d0 \ub300\ud574\uc11c\ub9cc \ub098\uc624\ub3c4\ub85d \uc815\ub9ac. \uc591\ucabd\uc744 \uac01\uac01 y, x\uc5d0 \ub300\ud574\uc11c \uc801\ubd84\ud558\uba74 \ub428. <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{matrix}\n\\begin{align*}\n\\frac{dy}{dx} = f(x)g(y) \\quad \\rightarrow\n\\end{align*}\n\\begin{align*}\n \\frac{dy}{g(y)} = f(x)dx \\\\\n\\int \\frac{1}{g(y)}dy = \\int f(x)dx\n\\end{align*}\n\\end{matrix}<\/pre><\/div>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p>\uc6081)<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{align*}\ny^{\\prime} = \\frac{x}{y} \\quad  \\rightarrow \\quad ydy = xdx \\\\\n\\int ydy = \\int xdx \\\\\n\\frac {1}{2}y^2 + C_1 = \\frac{1}{2}x^2 + C_2 \\\\\ny^2 = x^2 + C \\; or \\; y = \\pm \\sqrt{x^2 + C}\n\\end{align*}<\/pre><\/div>\n\n\n\n<p>\uc6082)<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{align*}\n\\frac{dy}{dx} - x = xy^2 \\\\\n\\int \\frac{1}{y^2+1}dy = \\int xdx \\\\\ntan^{-1} y + C_1= \\frac{1}{2}x^2 + C_2 \\\\\n\\therefore \\quad y = tan(\\frac{1}{2}x^2 + C)\n\n\\end{align*} \n\n<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"1st_order-integ-factor\">Linear 1st. order eq. : Intergrating factor<\/h2>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{align*}\n\\frac{dy}{dx} + p(x)y = q(x)\n\\end{align*} \n\n<\/pre><\/div>\n\n\n\n<p>Integrating factor \ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{gather*}\n\\frac{d(uy)}{dx} = uy\\prime + u\\prime y \\\\\n\\frac{du(x)}{dx} = up(x) \\quad \ub97c \\; \ub9cc\uc871\ud558\ub294 \\quad u \\quad \ub97c \\; \uac00\uc815\ud558\uba74, \\\\\n\\int \\frac{du}{u} = \\int pdx \\\\\nln|u| = \\int pdx \\\\\nu = e^{\\int pdx} \\quad : Integrating \\; factor!\n\\end{gather*}\n\n<\/pre><\/div>\n\n\n\n<p>\uc704\uc5d0\uc11c \uad6c\ud55c integrating factor\ub97c \uc6d0\ub798 \uc2dd\uc758 \uc591\ubcc0\uc5d0 \uacf1\ud558\uba74,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>uy^{\\prime} + upy = uq \\\\\n\\rightarrow  uy^{\\prime} + u^{\\prime} y = uq \\\\\n\\rightarrow d(uy)\/dx = uq<\/pre><\/div>\n\n\n\n<p>\uacb0\uacfc\uc801\uc73c\ub85c integrating factor\ub97c \uacf1\ud574\uc90c\uc73c\ub85c \ud574\uc11c, seperable \ud615\ud0dc\ub85c \ubc14\ub01c.<\/p>\n\n\n\n<p>\uc608 1)<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y^{\\prime} + y = e^x<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>integrating \\; factor \\;u = e^{\\int pdx} = e^x \\\\\ne^x(y^{\\prime} + y) = e^{2x} \\\\\n\\frac{d(e^x y)}{dx} =  ye^x = \\int e^{2x}dx \\\\\n= \\frac{1}{2}e^{2x} + C \\\\\n\\therefore y = \\frac{1}{2}e^x + Ce^{-x}<\/pre><\/div>\n\n\n\n<p>\uc608 2) RL circuit<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"181\" src=\"http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/03\/diff_eq01-300x181.png\" alt=\"\" class=\"wp-image-4271\" srcset=\"http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/03\/diff_eq01-300x181.png 300w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/03\/diff_eq01-768x463.png 768w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/03\/diff_eq01.png 995w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/figure>\n<\/div>\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>KVL : V - V_R - V_L = 0 \\\\\n \\begin{cases}\n   V_R = IR \\\\\n   V_L = L \\frac{dI}{dt}\n\\end{cases} <\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>L \\frac{dI}{dt} + RI = V<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>I^{\\prime} + \\frac{R}{L}I = \\frac{V}{L} \\\\\n\\begin{align*} \\end{align*} \\\\\nu = e^{\\int \\frac{R}{L}dt} = e^{\\frac{R}{L}t} \\\\\n\\begin{align*} \\end{align*} \\\\\n\\int e^{\\frac{Rt}{L}}[\\frac{dI}{dt} + \\frac{R}{L}I]dt = \\int \\frac{V}{L}e^{\\frac{Rt}{L}}dt \\\\\n\\begin{align*} \\end{align*} \\\\\ne^{\\frac{Rt}{L}}I = \\int \\frac{V}{L}e^{\\frac{Rt}{L}}dt = \\frac{V}{R}e^{\\frac{Rt}{L}} + C \\\\\n\\begin{align*} \\end{align*} \\\\\n I = \\frac{V}{R} + Ce^{-Rt\/L} \\\\\n\\begin{align*} \\end{align*} \\\\\nI(0) = I_0, I_0 = \\frac{V}{R} + C, C = I_0 - \\frac{V}{R} \\\\\n\\begin{align*} \\end{align*} \\\\\nI(t) = \\frac{V}{R} + (I_0 - \\frac{V}{R})e^{-Rt\/L}\n\n<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"homogeneous-1st-order-de\">Homogeneous 1st order D.E.<\/h2>\n\n\n\n<p>Not Separable, Non linear \ud55c \uacbd\uc6b0 Substitution Methods \ub97c \uc0ac\uc6a9. 1) Homogeneous, 2) Bernoulli <br>\uc5ec\uae30\uc11c homogeneous\ub294 linear equation\uc744 \ub9d0\ud558\ub294\uac8c \uc544\ub2d8.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\text{If the eq. is in the form,  } \\frac{dy}{dx} = f(x, y) \\\\\n\\begin{align*} \\end{align*} \\\\\n \\equiv M(x, y)dx + N(x, y)dy = 0\\\\\n\\begin{align*} \\end{align*} \\\\\n\\text{homogeneous condition : } f(x, y) = f(tx, ty)<\/pre><\/div>\n\n\n\n<p>\uc704\uc758 \uc870\uac74\uc744 \ub9cc\uc871\ud558\uba74, homogeneous. \uc774\uac74 homogeneous test \uc5d0 \uc0ac\uc6a9\ub420 \uc218 \uc788\uc74c.<\/p>\n\n\n\n<p>\uc774 \uacbd\uc6b0, substitution method\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uce58\ud658\ud558\ub294 \uac83.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{cases}\n\\quad y = vx \\quad or \\quad v = y\/x\\\\\n\\quad dy = vdx + xdv\n\\end{cases}<\/pre><\/div>\n\n\n\n<p>\uc774\ub97c \uc801\uc6a9\ud558\uba74, \uc6d0\ub798\uc2dd\uc774 seperable\uc774 \ub41c\ub2e4. <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\frac{dy}{dx} = f(x, y) \\\\\n\\begin{align*} \\end{align*} \\\\\n[ vdx + xdv ] = f(x, y)dx  = f(tx, ty)dx  \\\\\n\\begin{align*} \\end{align*} \\\\\nt = 1\/x \\;\\text{\ub85c \ub193\uc73c\uba74}, f(tx, ty) = f(1, y\/x) = f(y\/x) = f(v)\n\\begin{align*} \\end{align*} \\\\\n\\therefore \\frac{1}{f(v) + v}dv = \\frac{1}{x}dx<\/pre><\/div>\n\n\n\n<p>\uc608) <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\frac{dy}{dx} = \\frac{x}{y} + \\frac{y}{x} \\\\\n\\begin{align*} \\end{align*} \\\\\nf(tx, ty) = \\frac{tx}{ty} + \\frac{ty}{tx} = \\frac{x}{y} + \\frac{y}{x} = f(x, y) \\\\\n\\begin{align*} \\end{align*} \\\\\n\\therefore \\text{Homogeneous equation} \\\\\n\\begin{align*} \\end{align*} \\\\\ny = vx, dy = vdx+xdv \\\\\n\\begin{align*} \\end{align*} \\\\\n\\frac{vdx + xdv}{dx} = \\frac{x}{vx} + \\frac{vx}{x}\n\\begin{align*} \\end{align*} \\\\\nxdv = \\frac{1}{v}dx \\\\\n\\begin{align*} \\end{align*} \\\\\n\\int vdv = \\int \\frac{1}{x}dx \\\\\n\\begin{align*} \\end{align*} \\\\\n\\frac{1}{2}v^2 = ln|x| + C \\\\\n\\begin{align*} \\end{align*} \\\\\n\\frac{1}{2}(\\frac{y}{x})^2 = ln|x| + C \\\\\n\\begin{align*} \\end{align*} \\\\\ny^2 = 2x^2ln|x| + Cx^2 \\\\\n\\begin{align*} \\end{align*} \\\\\n\\therefore y = \\pm \\sqrt{2x^2ln|x| + Cx^2}<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"bernoui-de\">Bernoui D.E.<\/h2>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\frac{dy}{dx} + f(x)y = g(x)y^n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>v = y^{1-n} \\quad \\text{substitution} \\\\\n\\begin{align*} \\end{align*} \\\\\nn = 1 \\quad \\text{\uc774\uba74} \\quad \\text{linear} \\quad \\therefore n \\neq 0, 1 \\text{\uc778 \uacbd\uc6b0,} \\\\\n\\begin{align*} \\end{align*} \\\\\n\\text{substitution\uc744 \uc774\uc6a9\ud558\uba74 linear\ud615\ud0dc\ub85c \ubcc0\ud658 \uac00\ub2a5}<\/pre><\/div>\n\n\n\n<p>\uc608) <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\frac{dy}{dx} - \\frac{y}{x} = xy^2 <\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>v = 1\/y, \\frac{dv}{dx} = - \\frac{1}{y^2} \\frac{dy}{dx} \\quad \\text{substitution\uc744 \uc801\uc6a9} \\\\\n\\begin{align*} \\end{align*} \\\\\n\\frac{dv}{dx} + \\frac{1}{x}v = -x \\\\\n\\begin{align*} \\end{align*} \\\\\n\\text{integrating factor : } e^{\\int \\frac{1}{x} \\cdot dx} = e^{lnx} = x \\\\\n\\begin{align*} \\end{align*} \\\\\n\\int [x \\frac{dv}{dx} + v]dx = - \\int x^2 dx \\\\\n\\begin{align*} \\end{align*} \\\\\nxv = -\\frac{1}{3}x^3 + C \\\\\n\\begin{align*} \\end{align*} \\\\\nv = -\\frac{1}{3}x^2 + \\frac{C}{x} \\\\\n\\begin{align*} \\end{align*} \\\\\n\\frac{1}{y} = -\\frac{1}{3}x^2 + \\frac{1}{x}C = \\frac{-x^3 +3C}{3x}, 3C -&gt; C \\\\\n\\begin{align*} \\end{align*} \\\\\n\\therefore y = \\frac{3x}{C - x^3}\n<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"exact-de\">Exact D.E.<\/h2>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>M(x, y)dx + N(x, y)dy = 0 \\\\\n\\begin{align*} \\end{align*} \\\\\n\\text{is exact if there exists a continuously differentiable function } f, \\\\\n\\begin{align*} \\end{align*} \\\\\n\\frac{\\partial f}{\\partial x}  = M, \\frac{\\partial f}{\\partial y} = N<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>f(x, y) \\text{\uac00 \uc5f0\uc18d\uc778 \ud3b8\ub3c4\ud568\uc218\ub97c \uac00\uc9c0\uba74,} \\\\\n\\begin{align*} \\end{align*} \\\\\ndf =  \\frac{\\partial f}{\\partial x}\\cdot{dx} + \\frac{\\partial f}{\\partial y}\\cdot{dy} \\\\\n\\begin{align*} \\end{align*} \\\\\nf(x, y) = C \\quad \\text{\ub77c\uba74, chain rule\uc5d0 \uc758\ud574(\ub610\ub294 \uc704\uc758 df\uc0ac\uc6a9),} \\\\\n\\begin{align*} \\end{align*} \\\\\n\\frac{df}{dx} = \\frac{\\partial f}{\\partial x}\\cdot\\frac{dx}{dx} + \\frac{\\partial f}{\\partial y}\\cdot\\frac{dy}{dx} = 0 \\\\\n\\begin{align*} \\end{align*} \\\\ \nf_x + f_y \\cdot \\frac{dy}{dx} = 0 \\\\\n\\begin{align*} \\end{align*} \\\\ \n\\frac{dy}{dx} = - \\frac{f_x}{f_y} \\\\\n\\begin{align*} \\end{align*} \\\\ \n\\therefore \\text{\uc774\uc640 \uac19\uc740 \ud615\ud0dc\uc758 \ubbf8\ubd84\ubc29\uc815\uc2dd\uc758 \ud574\ub294} \\; f(x, y)= C \\; \\text{\uac00 \ub41c\ub2e4.}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\nabla f(x, y) = &lt; f_x, f_y &gt; \\\\\n\\begin{align*} \\end{align*} \\\\ \n\\text{vector field : } \\vec F = &lt; M, N &gt; \\\\ \n\\begin{align*} \\end{align*} \\\\\n\\vec F \\text{\ub294 gradient vector field = Slope field}<\/pre><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Exactness test \uc870\uac74 :<\/li>\n<\/ul>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>M(x, y)dx + N(x, y)dy = 0 \\\\\n\\begin{align*} \\end{align*} \\\\ \n\\frac{\\partial f}{\\partial x}  = M, \\frac{\\partial f}{\\partial y} = N \\\\\n\\begin{align*} \\end{align*} \\\\ \n\\text{\ub97c \ub9cc\uc871\ud558\ub294} \\; f \\; \\text{\uac00 \uc874\uc7ac\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790}\\\\\n\\text{\uc774 \ud568\uc218\uac00 \uc5f0\uc18d\uc774\uace0, 1\ucc28 \ud3b8\ubbf8\ubd84 \ub610\ud55c \uc5f0\uc18d\uc77c \ub54c,}\\\\\n\\begin{align*} \\end{align*} \\\\ \nf_{xy} = f_{yx} \\\\\n\\begin{align*} \\end{align*} \\\\ \n M_y = N_x \\\\\n\\begin{align*} \\end{align*} \\\\ \n\\text{\ubc18\ub300\ub85c } M_y \\neq N_x \\quad \\text{\uc774\uba74} \\quad f \\quad \\text{\ub294 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc74c.} \\\\\n\\begin{align*} \\end{align*} \\\\ \n\\therefore \\text{exactness test \uc870\uac74\uc774 \ub428.}\n<\/pre><\/div>\n\n\n\n<p>\uac04\ub2e8\ud788 \ub9d0\ud574, M\uc744 y\uc5d0 \ub300\ud574 \ubbf8\ubd84, N\uc744 x\uc5d0 \ub300\ud574 \ubbf8\ubd84\ud558\uc5ec \ub450 \uac12\uc774 \uac19\uc740\uc9c0 \ube44\uad50\ud574\ubcf4\uba74 exact D.E.\uc778\uc9c0 \ud655\uc778\uc774 \uac00\ub2a5\ud558\ub2e4. <\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\ud574\ub97c \uad6c\ud558\ub294 \ubc29\ubc95<\/li>\n<\/ul>\n\n\n\n<p>M, N\uc744 \uac01\uac01 x, y\uc5d0 \ub300\ud574 \uc801\ubd84\ud574\ubcf4\uba74 \ub458 \ub2e4 f(x, y)\uc774\ubbc0\ub85c \uc774 \uac12\uc744 \ube44\uad50\ud574\ubcf4\uba74 f(x, y)\ub97c \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\int M dx = \\int N dy = f(x, y)<\/pre><\/div>\n\n\n\n<p>\uc608) <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(x^2 + xy^2)dx + (x^2y - y^3)dy = 0<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>M_y = 2xy =  N_x \\\\\n\\begin{align*} \\end{align*} \\\\ \n\\therefore \\text{exact D.E.} \\\\\n\\begin{align*} \\end{align*} \\\\ \n\\int Mdx = \\int {x^2 + xy^2}dx = \\frac{1}{3}x^3 + \\frac{1}{2}x^2y^2 + C(y) \\\\\n\\begin{align*} \\end{align*} \\\\ \n\\int Ndy = \\int{x^2y-y^3}dy = \\frac{1}{2}x^2y^2 - \\frac{1}{4}y^4 + C(x) \\\\\n\\begin{align*} \\end{align*} \\\\ \n\\text{compare these two, } \\\\\n\\begin{align*} \\end{align*} \\\\ \nC(y) = -\\frac{1}{4}y^4 + C_1, C(x) = \\frac{1}{3}x^3 + C_2 \\\\\n\\begin{align*} \\end{align*} \\\\ \n\\therefore f(x, y) = \\frac{1}{3}x^3 + \\frac{1}{2}x^2y^2 -\\frac{1}{4}y^4 = C<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"linear-2nd-order-de\">Linear 2nd order D.E.<\/h2>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>ay^{\\prime\\prime}(x) + by^{\\prime}(x) + cy(x) = g(x)<\/pre><\/div>\n\n\n\n<p>g(x) = 0 \uc774\uba74, homogeneous equation.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>ay^{\\prime\\prime}(x) + by^{\\prime}(x) + cy(x) = 0<\/pre><\/div>\n\n\n\n<p>General solution for homogeneous eq. <\/p>\n\n\n\n<p>lenearity\uc5d0 \uc758\ud574, y1, y2\uac00 \ubc29\uc815\uc2dd\uc758 \ud574\ub77c\uba74 \uc5ec\uae30\uc5d0 \uc0c1\uc218\ub97c \uacf1\ud574\ub3c4 \uc5ec\uc804\ud788 \ud574\uc774\uace0, y1, y2\uc758 \uc870\ud569\ub3c4 \ud574\uac00\ub41c\ub2e4.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y_1, y_2 \\; \\text{\uac00 special solution \uc774\uace0, linearly independent \uc77c \ub54c,} \\\\\ny = C_1y_1 + C_2y_2<\/pre><\/div>\n\n\n\n<p>General solution for Non-homogeneous eq.<\/p>\n\n\n\n<p> \ubc29\uc815\uc2dd\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud2b9\uc218\ud574 <math data-latex=\"y_p\"><semantics><msub><mi>y<\/mi><mi>p<\/mi><\/msub><annotation encoding=\"application\/x-tex\">y_p<\/annotation><\/semantics><\/math> \ub97c \ucc3e\uc73c\uba74, homogeneous solution\uc5d0 \uc774\uac83\ub9cc \ub354\ud558\uba74 \uc77c\ubc18\ud574\uac00 \ub41c\ub2e4.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y = y_c + y_p, \\\\\n y_c : \\text{complementary solution, } \\\\\ny_p : \\text{particular solution } \\\\\n\\begin{align*} \\end{align*} \\\\ \n\\therefore y = C_1y_1 + C_2y_2 + y_p\n<\/pre><\/div>\n\n\n\n<p>general solution\uc744 \ubc29\uc815\uc2dd\uc5d0 \ub300\uc785\ud574\ubcf4\uba74, <math data-latex=\"y_c\"><semantics><msub><mi>y<\/mi><mi>c<\/mi><\/msub><annotation encoding=\"application\/x-tex\">y_c<\/annotation><\/semantics><\/math>  \ub294 0\uc774 \ub418\uace0, <math data-latex=\"y_p\"><semantics><msub><mi>y<\/mi><mi>p<\/mi><\/msub><annotation encoding=\"application\/x-tex\">y_p<\/annotation><\/semantics><\/math> \ub9cc \ub0a8\uc544 g(x)\uac00 \ub41c\ub2e4. <\/p>\n\n\n\n<p>\uc774\uc81c, \uc120\ud615 2\ucc28 \ubbf8\ubd84 \ubc29\uc815\uc2dd\uc758 \ud574\ub97c \uad6c\ud558\ub294 \ub2e4\uc591\ud55c \ubc29\ubc95\uc744 \ucf00\uc774\uc2a4 \ubc14\uc774 \ucf00\uc774\uc2a4\ub85c \uc54c\uc544\ubcf4\uc790.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"reduction-of-order\">2nd order D.E. : Reduction of Order.<\/h2>\n\n\n\n<p>\ud558\ub098\uc758 \ud574\ub97c \uc54c\uace0 \uc788\uc744 \ub54c, \ub098\uba38\uc9c0 \ud574\ub97c \uad6c\ud558\ub294 \ubc29\ubc95. \uc81c\ubaa9\ub300\ub85c \uce58\ud658\uc744 \ud574\uc11c 2\ucc28 \ubbf8\ubd84 \ubc29\uc815\uc2dd\uc744 1\ucc28 \ubbf8\ubd84 \ubc29\uc815\uc2dd\uc73c\ub85c \ub0ae\ucdb0\uc11c \ud478\ub294 \ubc29\ubc95\uc774\ub2e4. <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\text{\uc774\ubbf8 \uc54c\uace0 \uc788\ub294 \ud574\ub97c } \\; y_1 \\; \\text{\uc774\ub77c \ud558\uba74,} \\\\\n\\text{\ub098\uba38\uc9c0 \ud558\ub098\uc758 \ud574\ub294 } \\; y(x) = u(x)y_1(x) \\\\\n\\text{\uc774 \ud574\ub97c \ub300\uc785\ud574\uc11c u(x)\ub97c \uad6c\ud558\uba74 \ub41c\ub2e4.}<\/pre><\/div>\n\n\n\n<p>\uc774\uc640\uac19\uc774 u(x)\ub97c \uad6c\ud558\ub294 \uacfc\uc815\uc5d0\uc11c 2\ucc28 \ubbf8\ubd84 \ubc29\uc815\uc2dd\uc744 1\ucc28 \ubbf8\ubd84 \ubc29\uc815\uc2dd\uc73c\ub85c \uce58\ud658\ud558\ub294 \uacfc\uc815\uc774 \ucd94\uac00\ub85c \ub098\uc624\uac8c \ub428.<\/p>\n\n\n\n<p>\uc608)<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>x^2y^{\\prime\\prime} + 3xy^{\\prime} + y = 0, \\quad y_1 = \\frac{1}{x}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y = uy_1 \\; \\text{\uc774\ub77c\uace0 \ub193\uc73c\uba74,} \\; y = u\\cdot\\frac{1}{x} \\\\\n\\begin{align*} \\end{align*} \\\\ \ny^{\\prime} = u^{\\prime} \\frac{1}{x} + u(-\\frac{1}{x^2}), \\quad y^{\\prime\\prime} = u^{\\prime\\prime}\\frac{1}{x}  - u^{\\prime}\\frac{2}{x^2} - u(\\frac{2}{x^3}) \\\\\n\\begin{align*} \\end{align*} \\\\ \n\\text{\ubc29\uc815\uc2dd\uc5d0 \ub300\uc785\ud558\uba74,} \\\\\n\\begin{align*} \\end{align*} \\\\ \nx^2[u^{\\prime\\prime}\\frac{1}{x} - u^{\\prime}\\frac{2}{x^2} + u\\frac{2}{x^3}] + 3x[u^{\\prime}\\frac{1}{x} - u\\frac{1}{x^2}] + u\\frac{1}{x} = 0 \\\\\n\\begin{align*} \\end{align*} \\\\ \nxu^{\\prime\\prime} + u^{\\prime} = 0 \\\\\n\\begin{align*} \\end{align*} \\\\ \nv = u^{\\prime} \\; \\text{\uc73c\ub85c \uce58\ud658\ud558\uba74,} \\\\\n\\begin{align*} \\end{align*} \\\\ \nxv^{\\prime} + v = 0 \\\\\n\\begin{align*} \\end{align*} \\\\ \n \\text{seperable 1st order D.E\uc73c\ub85c v\ub97c \uad6c\ud560 \uc218 \uc788\ub2e4.}\\\\<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\frac{dv}{dx}x = -v \\\\\n\\begin{align*} \\end{align*} \\\\ \n-\\frac{1}{v}dv = \\frac{1}{x}dx \\\\\n\\begin{align*} \\end{align*} \\\\ \n\\ln v = -\\ln x + C_1 \\\\\n\\begin{align*} \\end{align*} \\\\ \nv = e^{-\\ln 1\/x}\\cdot e^{C_1}, \\; e^{C_1} \\rightarrow C_1 \\\\\n\\begin{align*} \\end{align*} \\\\ \nv = C_1e^{-\\ln 1\/x} =    \\frac{C_1}{x} \\\\\n\\begin{align*} \\end{align*} \\\\\n\\therefore u = C_1\\ln x + C_2 \\\\\n\\begin{align*} \\end{align*} \\\\\n\\text{general solution : } \\; y = uy_1 = C_1\\frac{\\ln x}{x} + \\frac{C_2}{x}<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Reduction of Order : Non-homogeneous<\/h2>\n\n\n\n<p>\uc608) <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y^{\\prime\\prime} - 5y^{\\prime} + 6y = e^{-x}, \\quad y_1 = e^{3x}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{aligned}\n\\text{let,} \\quad &amp;y = uy_1 = ue^{3x} \\\\\n&amp;y^{\\prime} = u^{\\prime} e^{3x} + 3ue^{3x} \\\\\n&amp;y^{\\prime\\prime}  = u^{\\prime\\prime} e^{3x} + 6u^{\\prime} e^{3x} + 9ue^{3x}\n\\end{aligned}<\/pre><\/div>\n\n\n\n<p>\uc704 \uac12\ub4e4\uc744 \uc6d0\ub798 \uc2dd\uc5d0 \ub300\uc785\ud558\uba74,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{aligned}\n&amp;e^{3x}[u^{\\prime\\prime} + 6u^{\\prime} + 9u - 5(u^{\\prime} 3u) + 6u] &amp;= e^{-x} \\\\\n&amp;u^{\\prime\\prime} + u^{\\prime} &amp;= e^{-4x}\n\\end{aligned}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{aligned}\n\\text{let,} \\quad &amp;v = u^{\\prime} ,\\\\\n&amp;v^{\\prime} + v = e^{-4x} \\\\\n&amp;\\frac{dv}{dx} + v -e^{-4x} = 0 \\\\\n\\text{integrating factor : } \\quad &amp;e^{\\int1\\cdot dx} = e^x ,\\\\\n&amp;e^x \\frac{dv}{dx} + e^xv - e^{-3x} = 0 \\\\\n&amp;[ve^x]^{\\prime} = e^{-3x} \\\\\n&amp;ve^x = \\int e^{-3x}dx = -\\frac{1}{3}e^{-3x} +C_1 \\\\\n&amp;v = -\\frac{1}{3}e^{-4x} + C_1e^{-x}  \\\\\nu = \\int vdx &amp;=\\frac{1}{12}e^{-4} + C_1e^{-x} + C_2 \\\\\n\\therefore y &amp;= \\frac{1}{12}e^{-x} + C_1e^{2x} + C_2e^{3x}  \n\\end{aligned}<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"constant-coefficients\">2nd order D.E. : Constant Coefficients <\/h2>\n\n\n\n<p>\ub2e4\uc74c\uacfc \uac19\uc774, coefficients \uac00 constant \uc77c \ub54c \ud480\uc774\ubc29\ubc95\uc774\ub2e4.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>ay^{\\prime\\prime} + by^{\\prime} +cy = 0 \\\\ \\text{a, b, c are constants}<\/pre><\/div>\n\n\n\n<p>\uc774\ub807\uac8c constant coefficients\ub97c \uac16\ub294 \uacbd\uc6b0\ub294 \ub2e4\uc74c\uc758 \ud574\ub97c \uc774\uc6a9\ud55c\ub2e4. <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{aligned}\n\\text{let, }\\quad &amp;y = e^{mx} \\\\\n&amp;y^{\\prime} = me^{mx} \\\\\n&amp;y^{\\prime\\prime} = m^2e^{mx} \n\\end{aligned}<\/pre><\/div>\n\n\n\n<p>\uc774 \uac12\ub4e4\uc744 \uc6d0\ub798 \ubc29\uc815\uc2dd\uc5d0 \ub123\uc5b4\ubcf4\uba74,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>e^{mx}(am^2 + bm + c) = 0 \\\\\n\\begin{align*} \\end{align*} \\\\\ne^{mx} \\;\\text{\ub294 0\uc774 \uc544\ub2c8\ubbc0\ub85c,} \\\\\n\\begin{align*} \\end{align*} \\\\\nam^2+bm+c = 0\\\\\n<\/pre><\/div>\n\n\n\n<p>coefficients\ub4e4\uc758 \ub2e8\uc21c 2\ucc28\ubc29\uc815\uc2dd\uc744 \ud480\uba74 \ud574\ub97c \uad6c\ud560 \uc218 \uc788\ub2e4. \uc774 2\ucc28\ubc29\uc815\uc2dd\uc744 characteristic polynomial form \uc774\ub77c\uace0 \ubd80\ub978\ub2e4. 2\ucc28\ubc29\uc815\uc2dd\uc758 \ud574\uac00 \uc2e4\uc218\uc778 2\uac1c\uc778 \uacbd\uc6b0\ub294 \ubb38\uc81c\uac00 \uc5c6\ub294\ub370, \uc911\ubcf5\ud574\ub85c \ud558\ub098\ub9cc \uc874\uc7ac\ud558\ub294 \uacbd\uc6b0, \ud5c8\uc218\uc758 \ud574\uac00 \uc874\uc7ac\ud558\ub294 \uacbd\uc6b0\uac00 \ubb38\uc81c\ub2e4. \uac01 \ucf00\uc774\uc2a4\ubcc4\ub85c \uc5b4\ub5bb\uac8c \ud480\uc5b4\uc57c \ud558\ub294\uc9c0 \ucc28\ucc28 \uc54c\uc544\ubcfc \uac83\uc774\ub2e4. <\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-268f00db5605e2a37ab3931042763225\" style=\"color:#005a8e\"> i) \ub450 \uac1c\uc758 \uc2e4\uc218 \ud574\ub97c \uac16\ub294 \uacbd\uc6b0,<\/p>\n\n\n\n<p>\uc608) <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y^{\\prime\\prime} + 3y^{\\prime} + 2y = 0<\/pre><\/div>\n\n\n\n<p>characteristic eq. <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{aligned}\nm^2 +3m + 2 = 0 \\\\\n(m+1)(m+2) = 0 \\\\\nm = -1, -2 \\\\\n\\therefore y = C_1e^{-x} + c_2e^{-2x}\n\\end{aligned}<\/pre><\/div>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-47e50ff88d7e897cfe73214ae008a645\" style=\"color:#005a8e\">ii) \uc911\ubcf5\ud574, \ud558\ub098\uc758 \ud574 <math data-latex=\"y = e^{mx}\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><msup><mi>e<\/mi><mrow><mi>m<\/mi><mi>x<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">y = e^{mx}<\/annotation><\/semantics><\/math> \uc778 \uacbd\uc6b0, <math data-latex=\"y = xe^{mx}\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>x<\/mi><msup><mi>e<\/mi><mrow><mi>m<\/mi><mi>x<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">y = xe^{mx}<\/annotation><\/semantics><\/math> \ub97c \ub610 \ub2e4\ub978 \ud574\ub85c \ub193\uc73c\uba74 \ub41c\ub2e4.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{aligned}\n\\text{let, } \\; y &amp;= xe^{mx} \\\\\ny^{\\prime} &amp;= e^{mx} + mxe^{mx} \\\\\ny^{\\prime\\prime} &amp;= 2me^{mx} + m^2xe^{mx}\n\\end{aligned}<\/pre><\/div>\n\n\n\n<p>\uc704\uc758 \uac12\uc744 \ubc29\uc815\uc2dd\uc5d0 \ub300\uc785\ud558\uba74, <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>a[2me^{mx} + m^2e^{mx}] + b[e^{mx} + mxe^{mx}] + cxe^{mx} \\\\\ne^{mx}[2am + axm^2+b+bxm+cx] \\\\\ne^{mx}[x(am^2+bm+c) + (2am+b)] \\\\\nam^2 +bm +c=0 \\; \\text{\uc774\ubbc0\ub85c, }\\\\\ne^{mx}(2am+b) \\\\\n\\text{\ud574\uac00 \ud558\ub098\ub77c\ub294\uac74, }\\; m = \\frac{-b}{2a} \\; \\text{\uc774\uace0 \uc774\ub97c \ub300\uc785\ud558\uba74, \ubc29\uc815\uc2dd\uc774 0\uc774 \ub428\uc744 \uc54c \uc218 \uc788\ub2e4.}\\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\therefore y = xe^{mx} \\; \\text{is another solution}<\/pre><\/div>\n\n\n\n<p>\uc608) <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{aligned}\ny^{\\prime\\prime} + 2y^{\\prime} + y = 0 \\\\\nm^2 + 2m + 1 = 0, \\\\\n(m+1)^2 = 0, \\;m = -1 \\\\\ny = C_1e^{-x} + C_2xe^{-x}\n\\end{aligned}<\/pre><\/div>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-6543fa6c7463420e0363793f523f5704\" style=\"color:#005a8e\">iii) complex solution\uc744 \uac16\ub294 \uacbd\uc6b0: <math data-latex=\"m = \\alpha \\pm \\beta i\"><semantics><mrow><mi>m<\/mi><mo>=<\/mo><mi>\u03b1<\/mi><mo>\u00b1<\/mo><mi>\u03b2<\/mi><mi>i<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">m = \\alpha \\pm \\beta i<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p>\uc704\uc758 \ud574\ub85c\ubd80\ud130, \ub2e4\uc74c\uacfc \uac19\uc740 \uc77c\ubc18\ud574\ub97c \uac16\ub294\ub2e4. <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y = e^{\\alpha x}(C_1cos\\beta x + c_2sin\\beta x)<\/pre><\/div>\n\n\n\n<p>\uc99d\uba85)<\/p>\n\n\n\n<p>Euler&#8217;s Formula : <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{aligned}\ne^{\\beta i x} &amp;= cos\\beta x + isin\\beta x \\\\\ne^{-\\beta ix} &amp;= cos(-\\beta x) + isin(-\\beta x) = cos(\\beta x) - isin(\\beta x)\n\\end{aligned}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y = e^{(\\alpha \\pm \\beta i)x} \\\\\ny_1 = e^{\\alpha x}\\cdot e^{\\beta ix}, \\; y_2 = e^{\\alpha x}\\cdot e^{-\\beta ix} \\\\\ny_1 = e^{\\alpha x}[cos\\beta x + isin\\beta x], \\; y_2 = e^{\\alpha x}[cos\\beta x - isin\\beta x] \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\text{let, }\n\\begin{cases}\nk_1 = \\frac{1}{2}y_1 + \\frac{1}{2}y_2  = e^{\\alpha x}cos\\beta x\\\\\nk_2 = \\frac{1}{2i}y_1 + \\frac{1}{2i}y_2 = e^{\\alpha x}sin\\beta x\\\\\n\\end{cases} \\\\\nk_1, k_2 \\; \\text{\ub294}\\;  y_1, y_2\\;\\text{\uc758 linear combination \uc774\ubbc0\ub85c,} \\\\\n\\text{\uc0c8\ub85c\uc6b4 lineary independent solution \uc774\ub2e4.} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\therefore y = e^{\\alpha x}(C_1cos\\beta x + C_2sin\\beta x)<\/pre><\/div>\n\n\n\n<p>\uc608)<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y^{\\prime\\prime} - 2y^{\\prime} +5y = 0 \\\\\n\\begin{aligned} \\end{aligned} \\\\\nm = \\frac{2\\pm\\sqrt{4 - 4\\times 5}}{2} = 1\\pm2i \\\\\n\\therefore y = e^x(C_1cos2x + C_2sin2x)<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"undetermined-coefficients\">2nd order D.E. : Undetermined Coefficients<\/h2>\n\n\n\n<p>\uc55e\uc758 Constants coefficients \ub85c \ub2e4\ub8ec\uac74 homogeneous eq.\uc758 \uacbd\uc6b0\uc774\uace0, \uc5ec\uae30\uc120 non-homogeneous \uc778 \uacbd\uc6b0\ub97c \ub2e4\ub8ec\ub2e4.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>ay^{\\prime\\prime} + by^{\\prime} + cy = g(x)<\/pre><\/div>\n\n\n\n<p>\uc55e\uc758 constant coefficients \ubc29\ubc95\uc744 \uc774\uc6a9\ud574\uc11c homogeneous \uc778 \uacbd\uc6b0\uc758 \ud574\ub97c \uad6c\ud574\ub193\uace0 \uc5ec\uae30\uc5d0 \ud2b9\uc218\ud574\ub97c \uad6c\ud558\uba74 \uadf8\uac83\uc774 non-homogeneous \uc778 \uacbd\uc6b0\uc758 \uc77c\ubc18\ud574\uac00 \ub41c\ub2e4.<\/p>\n\n\n\n<p>\ud2b9\uc218\ud574 <math data-latex=\"y_p\"><semantics><msub><mi>y<\/mi><mi>p<\/mi><\/msub><annotation encoding=\"application\/x-tex\">y_p<\/annotation><\/semantics><\/math> \ub294 g(x)\uc758 \ud615\ud0dc\uc5d0 \ub530\ub77c \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{matrix}\ng(x) \\quad\\rightarrow &amp;y_p\\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\text{Constant} &amp;A \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\text{Linear Polynomial} &amp; Ax +B \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\text{Quadratic Polynomial} &amp; Ax^2 + Bx + C \\\\\n\\begin{aligned} \\end{aligned} \\\\\nsin\\alpha x \\;\\text{or} \\; cos\\alpha x  &amp;Acos\\alpha x +Bsin\\alpha x \\\\\n\\begin{aligned} \\end{aligned} \\\\\ne^{\\alpha x} &amp;Ae^{\\alpha x}\n\\end{matrix}<\/pre><\/div>\n\n\n\n<p>\uc608)<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y^{\\prime\\prime} + 3y^{\\prime} + 2y = 8<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\text{i) homogeneous solution(constant coefficient)} \\\\\nm^2 + 3m + 2 = 0 \\\\\n(m+1)(m+2) = 0, \\; m = -1, -2 \\\\\ny_c = C_1e^{-x} + C_2e^{-2x} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\text{ii)} g(x) \\;\\text{is constant form} \\\\\ny_p = C \\\\\n0 + 3\\cdot0 +2\\cdot C = 8 \\\\\n\\therefore C = 4, \\; y_p = 4 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\text{general solution : }\\; y = y_c + y_p = C_1e^{-x} + C_2e^{-2x} + 4<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Exceptional case :  <math data-latex=\"y_p\"><semantics><msub><mi>y<\/mi><mi>p<\/mi><\/msub><annotation encoding=\"application\/x-tex\">y_p<\/annotation><\/semantics><\/math> is linearly dependent <math data-latex=\"y_c\"><semantics><msub><mi>y<\/mi><mi>c<\/mi><\/msub><annotation encoding=\"application\/x-tex\">y_c<\/annotation><\/semantics><\/math><\/h2>\n\n\n\n<p> <math data-latex=\"y_p\"><semantics><msub><mi>y<\/mi><mi>p<\/mi><\/msub><annotation encoding=\"application\/x-tex\">y_p<\/annotation><\/semantics><\/math>\uac00 <math data-latex=\"y_c\"><semantics><msub><mi>y<\/mi><mi>c<\/mi><\/msub><annotation encoding=\"application\/x-tex\">y_c<\/annotation><\/semantics><\/math> \uc5d0 linearly dependent \ud558\ub2e4\uba74 \uc774 <math data-latex=\"y_p \"><semantics><msub><mi>y<\/mi><mi>p<\/mi><\/msub><annotation encoding=\"application\/x-tex\">y_p <\/annotation><\/semantics><\/math> \ub294 \ud2b9\uc218\ud574\ub97c \ub9cc\uc871\uc2dc\ud0a4\uc9c0 \ubabb\ud574 \uc4f8 \uc218 \uc5c6\ub2e4. \uc774\ub7f4 \ub550, \uae30\uc874\uc5d0 \uc0ac\uc6a9\ud558\ub824\uace0 \ud588\ub358 <math data-latex=\"Ae^{\\alpha x}\"><semantics><mrow><mi>A<\/mi><msup><mi>e<\/mi><mrow><mi>\u03b1<\/mi><mi>x<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">Ae^{\\alpha x}<\/annotation><\/semantics><\/math> \uc5d0 constant coefficients\uc5d0\uc11c \uc911\ubcf5\ud574\uc758 \uacbd\uc6b0\ucc98\ub7fc x\ub97c \uacf1\ud574 \uc0ac\uc6a9\ud55c\ub2e4. \uc911\ubcf5\ub418\ub294 \ud574\ub97c \uc784\uc758\ub85c <math data-latex=\"y_1\"><semantics><msub><mi>y<\/mi><mn>1<\/mn><\/msub><annotation encoding=\"application\/x-tex\">y_1<\/annotation><\/semantics><\/math> \uc774\ub77c\uace0\ud558\uba74 \uc774 \ud574\ub294 <math data-latex=\"e^{mx}\"><semantics><msup><mi>e<\/mi><mrow><mi>m<\/mi><mi>x<\/mi><\/mrow><\/msup><annotation encoding=\"application\/x-tex\">e^{mx}<\/annotation><\/semantics><\/math> \ud615\ud0dc\uc774\ubbc0\ub85c,  <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y_p =  A\\cdot x\\cdot y_1 = A\\cdot x\\cdot e^{mx}<\/pre><\/div>\n\n\n\n<p>\uc774 \uac12\uc744 \ubc29\uc815\uc2dd\uc5d0 \uc801\uc6a9\ud558\uae30 \uc704\ud574 \ubbf8\ubd84\ud574\ubcf4\uba74,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{aligned}\ny_p &amp;= A\\cdot xy_1 \\\\\ny_p^{\\prime} &amp;= A\\cdot(xy_1)^{\\prime} =A\\cdot y_1 + A\\cdot xy_1^{\\prime}\\\\\ny_p^{\\prime\\prime} &amp;= A\\cdot(xy_1)^{\\prime\\prime} = A\\cdot 2y_1^{\\prime} + A\\cdot xy_1^{\\prime\\prime}\n\\end{aligned}<\/pre><\/div>\n\n\n\n<p>\uc774\ub97c \ubc29\uc815\uc2dd\uc5d0 \ub300\uc785\ud558\uba74,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>A\\cdot [a(xy_1^{\\prime\\prime} + 2y_1^{\\prime}) + b(xy_1^{\\prime} + y_1) + cxy_1] = g(x) \\\\\nA\\cdot [x(ay_1^{\\prime\\prime} + by_1^{\\prime} + cy_1) + 2ay_1^{\\prime} + by_1] = g(x) \\\\\nay_1^{\\prime\\prime} + by_1^{\\prime} + cy_1 = 0 \\; \\text{\uc774\ubbc0\ub85c, } \\\\\nA\\cdot [2ay_1^{\\prime} + by_1] = g(x) \\\\<\/pre><\/div>\n\n\n\n<p>\ud2b9\uc218\ud574\uac00 \uc911\ubcf5\ub418\ub294 \uacbd\uc6b0\ub294 g(x) \uac00 <math data-latex=\"e^{m x}\"><semantics><msup><mi>e<\/mi><mrow><mi>m<\/mi><mi>x<\/mi><\/mrow><\/msup><annotation encoding=\"application\/x-tex\">e^{m x}<\/annotation><\/semantics><\/math> \uc778 \uacbd\uc6b0\uc774\ubbc0\ub85c,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>A\\cdot (2a + b)e^{mx} = e^{m x} \\\\\n\\begin{aligned} \\end{aligned} \\\\\nA = \\frac{1}{2a+b}\n<\/pre><\/div>\n\n\n\n<p>\uc704\uc640\uac19\uc774 A\ub97c \uad6c\ud558\uba74, <math data-latex=\"A\\cdot x\\cdot e^{mx}\"><semantics><mrow><mi>A<\/mi><mo>\u22c5<\/mo><mi>x<\/mi><mo>\u22c5<\/mo><msup><mi>e<\/mi><mrow><mi>m<\/mi><mi>x<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">A\\cdot x\\cdot e^{mx}<\/annotation><\/semantics><\/math> \ub85c\ubd80\ud130  <math data-latex=\"y_p \"><semantics><msub><mi>y<\/mi><mi>p<\/mi><\/msub><annotation encoding=\"application\/x-tex\">y_p <\/annotation><\/semantics><\/math>  \ub97c \uad6c\ud560 \uc218 \uc788\ub2e4. <\/p>\n\n\n\n<p>\uc608)<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y^{\\prime\\prime} + 3y^{\\prime} +2y = e^{-2x}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>m^2 + 3m + 2 = 0 \\\\\n(m+1)(m+2) = 0 \\;, m = -1, -2 \\\\\ny_c = C_1e^{-x} + C_2e^{-2x} \\\\\ng(x) = e^{-2x}  \\rightarrow y_p = Ae^{-2x}<\/pre><\/div>\n\n\n\n<p>\ud2b9\uc218\ud574\uac00 \ubcf4\uc870\ud574\uc640 \uc911\ubcf5\ub41c\ub2e4. <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\text{let, }\\; y_p = Axe^{-2x} \\\\\ny_p^{\\prime} = A2^{-2x} + Ax(-2e^{-2x}) \\\\\ny_p^{\\prime\\prime} = -4Ae^{-2x} + 4Axe^{-2x} <\/pre><\/div>\n\n\n\n<p>\uc704 \uac12\uc744 \ub300\uc785\ud574\ubcf4\uba74,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>[-4Ae^{-2x} + 4Axe^{-2x}] + 3[Ae^{-2x} - 2Axe^{-2x}] + 2[Axe^{-2x}] = e^{-2x} \\\\\n-Ae^{-2x} = e^{-2x} \\\\\n\\therefore A = -1<\/pre><\/div>\n\n\n\n<p>\ud574\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y_p = -xe^{-2x} \\\\\ny = C_1e^{-x} + C_2e^{-2x} - xe^{-2x}<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"wronskian\">Wronskian For a Group of Functions<\/h2>\n\n\n\n<p>n\uac1c\uc758 \uc8fc\uc5b4\uc9c4 \ud568\uc218\uc5d0 \ub300\ud574 Wronskian\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>W = det\n\\begin{bmatrix}\ny_1 &amp;... &amp;y_n \\\\\ny_1^{\\prime} &amp;   &amp;y_n^{\\prime} \\\\\ny_1^{\\prime\\prime} &amp;\\ddots  &amp;y_n^{\\prime\\prime} \\\\\n\\vdots  &amp; &amp;\\vdots \\\\\ny_1^{(n-1)} &amp;  &amp;y_n^{(n-1)} \\\\\n\\end{bmatrix}<\/pre><\/div>\n\n\n\n<p>\uc815\ud655\ud55c \uc99d\uba85\uc740 \uc774\ud574\ub97c \ubabb\ud588\uc9c0\ub9cc, \ub300\ub7b5 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\ub9ac\uac00\ub2a5.<\/p>\n\n\n\n<p>n\uac1c\uc758 \ud568\uc218\ub4e4 <math data-latex=\"y_1, y_2, \\dots y_n\"><semantics><mrow><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>y<\/mi><mn>2<\/mn><\/msub><mo separator=\"true\">,<\/mo><mo>\u2026<\/mo><msub><mi>y<\/mi><mi>n<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">y_1, y_2, \\dots y_n<\/annotation><\/semantics><\/math> \uacfc \uc774\ub4e4\uc758 \ubbf8\ubd84\uc73c\ub85c \ub9cc\ub4e4\uc5b4\uc9c4 matrix A\uc640 n\uac1c\uc758 element <math data-latex=\"C_1, C_2, \\dots C_n \"><semantics><mrow><msub><mi>C<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>C<\/mi><mn>2<\/mn><\/msub><mo separator=\"true\">,<\/mo><mo>\u2026<\/mo><msub><mi>C<\/mi><mi>n<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">C_1, C_2, \\dots C_n <\/annotation><\/semantics><\/math> \uc73c\ub85c \uad6c\uc131\ub41c \ubca1\ud130 C\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc368\ubcf4\uc790.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{bmatrix}\ny_1 &amp;... &amp;y_n \\\\\ny_1^{\\prime} &amp;   &amp;y_n^{\\prime} \\\\\ny_1^{\\prime\\prime} &amp;\\ddots  &amp;y_n^{\\prime\\prime} \\\\\n\\vdots  &amp; &amp;\\vdots \\\\\ny_1^{(n-1)} &amp;  &amp;y_n^{(n-1)} \\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nC_1 \\\\\nC_2 \\\\\nC_3 \\\\\n\\vdots \\\\\nC_n\n\\end{bmatrix} = 0<\/pre><\/div>\n\n\n\n<p><\/p>\n\n\n\n<p>A\uac00 singular matrix( det(A) = 0 )\uc774\uba74, A\uc758 \uc5ed\ud589\ub82c\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>C_1y_1 + C_2y_2 + \\dots + C_ny_n = 0 \\\\\nC_1y_1^{\\prime} + C_2y_2^{\\prime} + \\dots + C_ny_n^{\\prime} = 0 \\\\\n\\vdots \\\\\nC_1y_1^{n-1} + C_2y_2^{n-1} + \\dots + C_ny_n^{n-1} = 0 \\\\<\/pre><\/div>\n\n\n\n<p>\uc989, <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>C_1y_1  = -(C_2y_2 + \\dots + C_ny_n)<\/pre><\/div>\n\n\n\n<p>\uc640 \uac19\uc774 n\uac1c\uc758 \ud568\uc218\ub4e4\uc740 \uc11c\ub85c \ub2e4\ub978 \ud568\uc218\uc5d0 \ub300\ud574 dependent\ud558\ub2e4. <\/p>\n\n\n\n<p>A\uc758 \uc5ed\ud589\ub82c\uc774 \uc874\uc7ac(det(A) <math data-latex=\"\\ne\"><semantics><mo lspace=\"0em\" rspace=\"0em\">\u2260<\/mo><annotation encoding=\"application\/x-tex\">\\ne<\/annotation><\/semantics><\/math> 0) \ud558\uba74, C\ub294 trivial solution zero vector \ub9cc \uac16\uac8c \ub418\ubbc0\ub85c n\uac1c\uc758 \ud568\uc218 <math data-latex=\"y_1, y_2, \\dots y_n\"><semantics><mrow><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>y<\/mi><mn>2<\/mn><\/msub><mo separator=\"true\">,<\/mo><mo>\u2026<\/mo><msub><mi>y<\/mi><mi>n<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">y_1, y_2, \\dots y_n<\/annotation><\/semantics><\/math> \ub294 \uc11c\ub85c linearly independent \ud558\ub2e4.     <\/p>\n\n\n\n<p>\ub2e4\uc2dc\ub9d0\ud574,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>W(x) \\ne 0 \\quad \\rightarrow y_1, y_2, \\dots, y_n \\quad \\text{are linearly independent}<\/pre><\/div>\n\n\n\n<p>\uc608)<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>e^{4x}, xe^{4x}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>W = \n\\begin{vmatrix}\ne^{4x} &amp;xe^{4x} \\\\\n4e^{4x} &amp;e^{4x} + 4xe^{4x} \\\\\n\\end{vmatrix}\n= e^{8x} \\\\\n\\begin{aligned}\n\\end{aligned} \\\\\n\\therefore e^{4x} \\quad \\text{and} \\quad xe^{4x} \\quad \\text{are linearly independent}<\/pre><\/div>\n\n\n\n<p> exponential\uc5d0 x\ub97c \uacf1\ud558\uba74 \uc11c\ub85c linearly independent \ud55c\uac78 \uc5ec\uae30\uc11c \ubcf4\uc5ec\uc8fc\uae30 \ub54c\ubb38\uc5d0. \uc55e\uc5d0\uc11c constant coefficient\uc758 \uc911\ubcf5\ud574\ub098 undetermined coefficent\uc5d0\uc11c exponential \ud615\ud0dc\uc758 \ud574\uac00 \uacb9\uce60 \ub54c, x\ub97c \uacf1\ud574\uc900\uac8c \uc5b4\ub5bb\uac8c \ub2e4\ub978 \ud574\uac00 \ub418\ub294\uc9c0 \uc5ec\uae30\uc11c \uc54c \uc218 \uc788\ub2e4.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"variation-of-parameters\">2nd order D.E. : Variation of Parameters<\/h2>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y^{\\prime\\prime} + P(x)y^{\\prime} + Q(x)y = g(x)<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y_c = C_1y_1 + C_2 y_2 \\quad \\text{\uc77c \ub54c,}\\\\\ny_p = u_1(x)y_1 + u_2(x)y_2 \\quad \\text{\ub77c\uace0 \ub193\uace0} \\; u_1, u_2 \\text{\ub97c \uad6c\ud558\ub294 \ubc29\ubc95.}\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{aligned}\ny_p &amp;= u_1y_1 + u_2y_2 \\\\\ny_p^{\\prime} &amp;= u_1^{\\prime} y_1 + u_1y_1^{\\prime} + u_2^{\\prime} y_2 + u_2y_2^{\\prime} \\\\\ny_p^{\\prime\\prime} &amp;= u_1^{\\prime\\prime} y_1 +  u_1^{\\prime} y_1^{\\prime} + u_1^{\\prime} y_1^{\\prime} + u_1y_1^{\\prime\\prime} + u_2^{\\prime\\prime} y_2 + u_2^{\\prime} y_2^{\\prime} + u_2^{\\prime} y_2^{\\prime} + u_2y_2^{\\prime\\prime} \\\\\n&amp;= u_1^{\\prime\\prime} y_1 +  2u_1^{\\prime} y_1^{\\prime} + u_1y_1^{\\prime\\prime} + u_2^{\\prime\\prime} y_2 + 2u_2^{\\prime} y_2^{\\prime} + u_2y_2^{\\prime\\prime}\n\\end{aligned}<\/pre><\/div>\n\n\n\n<p>\ubc29\uc815\uc2dd\uc5d0 \ub300\uc785\ud558\uba74,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{aligned}\n[u_1^{\\prime\\prime} y_1 +  2u_1^{\\prime} y_1^{\\prime} + u_1y_1^{\\prime\\prime} + u_2^{\\prime\\prime} y_2 + 2u_2\\prime y_2\\prime + u_2y_2^{\\prime\\prime}] + \\\\\nP(x)[u_1^{\\prime} y_1 + u_1y_1^{\\prime} + u_2^{\\prime} y_2 + u_2y_2^{\\prime}] + \\\\\nQ(x)[u_1y_1 + u_2y_2]  = g(x)  \\\\\n\\end{aligned} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\begin{aligned}\nu_1[y_1^{\\prime\\prime} + P(x)y_1^{\\prime} + Q(x)y_1] +\\\\\n u_2[y_2^{\\prime\\prime} + P(x)y_2^{\\prime} + Q(x)y_2] + \\\\\n(y_1u^{\\prime\\prime} + y_1^{\\prime} u_1^{\\prime}) + (y_2u_2^{\\prime\\prime} + y_2^{\\prime} u_2^{\\prime}) + P(x)(y_1u_1^{\\prime} + y_2u_2^{\\prime}) + y_1^{\\prime} u_1^{\\prime} + y_2^{\\prime} u_2^{\\prime} = g(x)\n\\end{aligned} \\\\<\/pre><\/div>\n\n\n\n<p>\uc55e\uc758 \ub450 \ud56d\uc740 homogeneous eq. \uc758 \ud574 \uc774\ubbc0\ub85c 0\uc774 \ub428. \ub9c8\uc9c0\ub9c9 \ud56d\uc740 \uad04\ud638\ub85c \ubb36\uc740 \ubd80\ubd84\uc744 \uc815\ub9ac\ud574\ubcf4\uba74,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\frac{d}{dx}(y_1u_1^{\\prime} + y_2u_2^{\\prime}) + P(x)(y_1u_1^{\\prime} + y_2u_2^{\\prime}) + y_1^{\\prime} u_1^{\\prime} + y_2^{\\prime} u_2^{\\prime} = g(x) \\\\\ny_1u_1^{\\prime} + y_2u_2^{\\prime} = 0 \\quad \\text{\uc744 \ub9cc\uc871\ud55c\ub2e4\uace0 \uac00\uc815(\uc77c\uc885\uc758 trick),} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\begin{cases}\n&amp;y_1u_1^{\\prime} + y_2u_2^{\\prime} = 0 \\\\\n&amp;y_1^{\\prime} u_1^{\\prime} + y_2^{\\prime} u_2^{\\prime} = g(x)\\\\\n\\end{cases} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\rArr \\begin{bmatrix}\ny_1 &amp;y_2 \\\\\ny_1^{\\prime} &amp;y_2^{\\prime} \\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nu_1^{\\prime} \\\\\nu_2^{\\prime}  \\\\\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n0 \\\\\ng(x)  \\\\\n\\end{bmatrix} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\text{\ud06c\ub798\uba38 \uacf5\uc2dd\uc73c\ub85c \ubd80\ud130,} \\\\\n\\begin{aligned} \\end{aligned} \\\\\nu_1^{\\prime} = \\frac{W_1}{W}, \\quad W_1 = \n\\begin{vmatrix}\n0 &amp;y_2 \\\\\ng(x) &amp;y_2^{\\prime}\n\\end{vmatrix} \\\\\n\\begin{aligned} \\end{aligned} \\\\\nu_2^{\\prime} = \\frac{W_2}{W}, \\quad W_2 = \n\\begin{vmatrix}\ny_1 &amp;0 \\\\\ny_1^{\\prime} &amp;g(x)\n\\end{vmatrix} \\\\<\/pre><\/div>\n\n\n\n<p>\uc704\uc758 \uac12\uc744 \uad6c\ud574\uc11c \uc801\ubd84\ud558\uba74 <math data-latex=\"u_1, u_2\"><semantics><mrow><msub><mi>u<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>u<\/mi><mn>2<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">u_1, u_2<\/annotation><\/semantics><\/math>   \ub97c \uad6c\ud560 \uc218 \uc788\uace0, \uc774\ub85c\ubd80\ud130 \ud574\ub97c \uad6c\ud560 \uc218 \uc788\ub2e4. \ud574\uac00 \uad6c\ud574\uc9c4\ub2e4\uba74, \uc911\uac04\uc5d0 \ud588\ub358 \uac00\uc815\uc740 \ub9cc\uc871\ud558\uba70 \ubcc4 \ubb38\uc81c\uac00 \ub418\uc9c0 \uc54a\ub294\ub2e4. <\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"cauchy-euler-eq\">Cauchy-Euler Eq.<\/h2>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>ax^2y^{\\prime\\prime} +bxy^{\\prime} + cy = 0<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{aligned}\n\\text{let, }\\quad y &amp;= x^m \\\\\ny^{\\prime} &amp;= mx^{m-1} \\\\\ny^{\\prime\\prime} &amp;= m(m-1)x^{m-2} \\\\\n\\end{aligned}<\/pre><\/div>\n\n\n\n<p>\uc704 \uac12\uc744 \ubc29\uc815\uc2dd\uc5d0 \ub300\uc785\ud558\uba74,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{aligned}\nax^2[m(m-1)x^{m-2}] + bxmx^{m-1} + cx^m &amp;= 0 \\\\\nx^m[am(m-1) + bm + c] &amp;= 0 \\\\\nam^2 + (b-a)m + c &amp;= 0 \\quad \\text{solve this,} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\text{general solution : }  \\quad y = c_1x^{m1} + c_2x^{m2}\n\\end{aligned}<\/pre><\/div>\n\n\n\n<p>\uc608)<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>x^2y^{\\prime\\prime} - 3xy^{\\prime} + 3y = 0 \\\\\n\\begin{aligned} \\end{aligned} \\\\\nm^2 + (-3-1)m + 3 = 0\\\\\n(m-1)(m-3) = 0 \\;, \\; m = 1, 3 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\therefore y_1 = x, y_2 = x^3 \\\\\n\ny = c_1x + c_2x^3<\/pre><\/div>\n\n\n\n<p>constant coefficients \uc758 \uacbd\uc6b0\uc640 \ub9c8\ucc2c\uac00\uc9c0 \ubb38\uc81c\uac00 \uc788\ub2e4. \uc911\ubcf5\ud574\uc640 complex number\uc778 \uacbd\uc6b0.<\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-e0cd62887e7f8d174a9088a24d726a60\" style=\"color:#065785\">i) \ub450 \uac1c\uc758 \uc2e4\uc218 \ud574\uc778\uacbd\uc6b0,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y = C_1x^{m1} +C_2x^{m2}<\/pre><\/div>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-82f920bc32bbab739210d931a40f9412\" style=\"color:#065785\">ii) \uc911\ubcf5\ud574\uc778 \uacbd\uc6b0,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y_1 = x^{m1}, y_2 = y_1\\cdot ln|x| = x^{m1}\\cdot ln|x| \\\\\n\\begin{aligned} \\end{aligned} \\\\\ny = C_1x^{m1} + c_2x^{m1}ln|x|<\/pre><\/div>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-72dda3b4bf1e0e9e90d0b3c6783a1f16\" style=\"color:#065785\">iii) complex number\uac00 \ud574\uc778 \uacbd\uc6b0, Constant Coefficients \ub54c\uc640 \uc720\uc0ac\ud558\ub2e4.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>m = \\alpha \\pm \\beta i \\\\\n\\begin{aligned} \\end{aligned} \\\\\ny = C_1x^\\alpha cos(\\beta ln|x|) + C_2x^\\alpha sin(\\beta ln|x|)<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Cauchy-Euler Eq. Non-Homogeneous<\/h2>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>ax^2y^{\\prime\\prime} + bxy^{\\prime} + cy = g(x)<\/pre><\/div>\n\n\n\n<p>i) \uc704\uc758 \ubc29\ubc95\uc73c\ub85c homogeneous solution <math data-latex=\"y_c\"><semantics><msub><mi>y<\/mi><mi>c<\/mi><\/msub><annotation encoding=\"application\/x-tex\">y_c<\/annotation><\/semantics><\/math>  \ub97c \uad6c\ud568.<\/p>\n\n\n\n<p>ii) <math data-latex=\"ax^2\"><semantics><mrow><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">ax^2<\/annotation><\/semantics><\/math> \uc73c\ub85c \ub098\ub204\uace0, variation of Parameters \uc774\uc6a9.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>W = \\begin{vmatrix}\ny_1 &amp;y_2 \\\\\ny_1^{\\prime} &amp;y_2^{\\prime}\n\\end{vmatrix},\nW_1 = \\begin{vmatrix}\n0 &amp;y_2 \\\\\ng\/ax^2 &amp;y_2^{\\prime}\n\\end{vmatrix},\nW_2 = \\begin{vmatrix}\ny_1 &amp;0 \\\\\ny_1^{\\prime} &amp;g\/ax^2\n\\end{vmatrix} \\\\\n\\begin{aligned} \\end{aligned} \\\\\nu_1^{\\prime} = \\frac{W1}{W}, u_2^{\\prime} = \\frac{W_2}{W} \\\\\n\\begin{aligned} \\end{aligned} \\\\\ny_p = u_1y_1 +u_2y_2 \\\\\n\\begin{aligned} \\end{aligned} \\\\\ny = y_c + y_p<\/pre><\/div>\n\n\n\n<p>\uc608)<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>x^2y^{\\prime\\prime} - 2xy^{\\prime} -4y = \\frac{1}{x}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>m^2 - 3m - 4 = 0 \\\\\n(m-4)(m+1) = 0 , \\quad m = 4, 1 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\therefore y_c = C_1x^4 + \\frac{C_2}{x}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\frac{g}{ax^2} = x^{-3} \\\\\n\\begin{aligned} \\end{aligned} \\\\\nW = \\begin{vmatrix}\nx^4 &amp;x^{-1} \\\\\n4x^3 &amp;-x^{-2}\n\\end{vmatrix} = -x^2 - 4x^2 = -5x^2 \\\\\n\\begin{aligned} \\end{aligned} \\\\\nW_1 = \\begin{vmatrix}\n0 &amp;x^{-1} \\\\\nx^{-3} &amp;-x^{-2}\n\\end{vmatrix} = -x^{-4} \\\\\n\\begin{aligned} \\end{aligned} \\\\\nW_2 = \\begin{vmatrix}\nx^4 &amp;0 \\\\\n4x^3 &amp;x^{-3}\n\\end{vmatrix} = x \\\\\n\\begin{aligned} \\end{aligned} \\\\\nu_1^{\\prime} = \\frac{-x^{-4}}{-5x^2} = \\frac{1}{5}x^{-6}, \\quad u_2^{\\prime} =\\frac{x}{-5x^2} = -\\frac{1}{5}x^{-1} \\\\\n\\begin{aligned} \\end{aligned} \\\\\nu_1 = -\\frac{1}{25}x^{-5}, \\quad u_2 = -\\frac{1}{5}ln|x| \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\therefore y_p = \\frac{-x^4}{25x^5} + (-\\frac{1}{5})\\cdot ln|x|\\cdot\\frac{1}{x} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n= -\\frac{1}{25x} - \\frac{1}{5x}ln|x| <\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y = y_c + y_p = C_1x^4 + C_2x^{-1} -\\frac{1}{25}x^{-1} - \\frac{ln|x|}{5x} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n= C_1x^4 + C_2\\frac{1}{x} - \\frac{ln|x|}{5x}<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"laplace-transform\">Laplace Transform<\/h2>\n\n\n\n<p>\uc815\uc758<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\mathcal{L}\\{f(t)\\} = \\int_0^\\infin f(t)e^{-st}dt<\/pre><\/div>\n\n\n\n<p>laplace transform\uc740 t\uc758 \ud568\uc218\ub97c t\uc5d0 \ub300\ud574 \uc801\ubd84\ud558\uc5ec t\ub97c \uc5c6\uc560\uace0 \ub300\uc2e0, exponential \ud615\ud0dc\uc758 s\uc758 \ud568\uc218\ub85c \ubc14\uafd4\uc8fc\ub294 \ubcc0\ud658\uc774\ub2e4. \uc801\ubd84\ud615\ud0dc\ub97c \ubcf4\uba74, f(t)\uc5d0 exponentialy decrease\ud558\ub294 \ud568\uc218\ub97c \uacf1\ud558\ubbc0\ub85c, f(t)\uac00 exponential \ubcf4\ub2e4 \ud06c\uac8c \ubc1c\uc0b0\ud558\uc9c0 \uc54a\ub294 \uc774\uc0c1, \uac12\uc740 \uc218\ub834\ud55c\ub2e4. \ubc1c\uc0b0\ud558\ub294 \uacbd\uc6b0\ub294 \ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658\uc73c\ub85c \ub2e4\ub8f0 \uc218 \uc5c6\uc9c0\ub9cc, \uc790\uc5f0\uacc4\uc5d0\uc120 \ub300\ubd80\ubd84 \uc218\ub834\ud558\uae30 \ub54c\ubb38\uc5d0 \uc720\ud6a8\ud558\ub2e4. <\/p>\n\n\n\n<p>\ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658\uc744 \ubbf8\ubd84\ubc29\uc815\uc2dd\uc5d0\uc11c \uc0ac\uc6a9\ud558\ub294 \ubc29\ubc95\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"640\" height=\"507\" src=\"http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/laplace_transform.png\" alt=\"\" class=\"wp-image-4514\" srcset=\"http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/laplace_transform.png 640w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/laplace_transform-300x238.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/figure>\n<\/div>\n\n\n<p>\ubbf8\ubd84\ubc29\uc815\uc2dd\uc744 \ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658\uc73c\ub85c \ubcc0\ud615\ud558\uba74, \ubcf4\ub2e4 \uac04\ub2e8\ud558\uac8c \ud480\uc774\uac00 \uac00\ub2a5\ud55c s\uc5d0 \ub300\ud55c Algebraic expression\ud615\ud0dc\uac00 \ub41c\ub2e4. \uc5ec\uae30\uc11c s\uc5d0 \ub300\ud55c \ud574\ub97c \uad6c\ud55c \ud6c4, \ub77c\ud50c\ub77c\uc2a4 \uc5ed\ubcc0\ud658\uc744 \uc2dc\ucf1c\uc8fc\uba74 \uc6d0\ub798 \uad6c\ud558\uace0\uc790 \ud588\ub358 \ubbf8\ubd84\ubc29\uc815\uc2dd\uc758 \ud574\uac00 \ub41c\ub2e4. \ubc29\uc815\uc2dd\uc744 t\uc5d0 \ub300\ud55c space\uc5d0\uc11c s\uc5d0 \ub300\ud55c space\ub85c \ubcc0\ud658 \ud6c4, s\ub97c \uc27d\uac8c \uad6c\ud55c \ud6c4\uc5d0 \uc774\uac78 \ub2e4\uc2dc t\uc5d0 \ub300\ud55c space\ub85c \ub418\ub3cc\ub9ac\ub294 \ubc29\uc2dd\uc774\ub2e4. <\/p>\n\n\n\n<p>\uae30\ud638\ub85c\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \ud45c\uc2dc\ud55c\ub2e4.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>f(t) \\quad \\rightarrow \\mathcal{L}\\{ f(t) \\} \\rightarrow F(s) \\\\\n\\begin{aligned} \\end{aligned}\\\\\nf(t) \\quad \\leftarrow \\mathcal{L}^{-1}\\{ f(t) \\} \\leftarrow F(s)<\/pre><\/div>\n\n\n\n<p>f(t) \ub294 \uc6d0\ub798\uc758 \ud568\uc218\uc774\uace0 F(s) \ub294 \ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658\ub41c \ud568\uc218\uc774\ub2e4. <\/p>\n\n\n\n<p>\ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658\uc740 \ud2b9\uc815 \ud615\ud0dc\uc758 \ubcc0\ud658\uc744 \ubbf8\ub9ac \uacc4\uc0b0\ud574\ub193\uace0 \uc0ac\uc6a9\ud558\ub294\uac8c \ud3b8\ud558\ub2e4. \uc5ec\uae30\uc11c \ub2e4\ub8f0 \ud615\ud0dc\ub294 \ub2e4\uc74c\uc758 6\uac00\uc9c0\uc774\ub2e4.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{array}{ll}\n\\textcircled{1}\\;\\mathcal{L}\\{1\\} &amp;\\textcircled{4}\\; \\mathcal{L}\\{e^{at}\\} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\textcircled{2}\\;\\mathcal{L}\\{t\\} &amp; \\textcircled{5}\\;\\mathcal{L}\\{\\sin(kt)\\} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\textcircled{3}\\;\\mathcal{L}\\{t^n\\} &amp; \\textcircled{6}\\;\\mathcal{L}\\{\\sinh(at)\\} \\\\\n\\end{array}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\textcircled{1}\\;\\mathcal{L}\\{1\\} &amp;= \\int_0^\\infin (1)\\cdot e^{-st}dt = \\lim\\limits_{b \\to \\infin}\\int_0^b\\ e^{-st}dt \\\\\n&amp;=\\lim\\limits_{b \\to \\infin}[-\\frac{1}{s}\\cdot e^{-st}\\rbrack_0^b = \\lim\\limits_{b \\to \\infin}[-\\frac{1}{s}\\cdot e^{-sb} -\\frac{1}{s}\\cdot e^{0}] \\\\\n&amp; b \\to \\infin \\Rightarrow  \\frac{1}{s}\\cdot e^{-sb} \\to 0 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n&amp;\\therefore  \\mathcal{L}\\{1\\} = \\frac{1}{s}\n\\end{split}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\textcircled{2}\\;\\mathcal{L}\\{t\\} &amp;= \\int_0^\\infin t\\cdot e^{-st}dt \\\\\n\\text{let,} \\; &amp;u = t, du = dt, dv = e^{-st}dt, v = -\\frac{1}{s}e^{-st}dt \\\\\n\\text{apply,}\\;&amp;\\int uv\\prime = \\int [(uv)\\prime - u\\prime v]\\\\\n&amp;= \\lim\\limits_{b \\to \\infin}[-\\frac{t}{s}e^{-st}]_0^b + \\lim\\limits_{b \\to \\infin}\\frac{1}{s}\\int_0^b e^{-st}dt \\\\\n&amp;= \\lim\\limits_{b \\to \\infin} [-\\frac{t}{s}e^{-st} - \\frac{1}{s^2}e^{-st}]_0^b \\\\\n&amp;= \\lim\\limits_{b \\to \\infin}[-\\frac{b}{s}e^{-sb} - \\frac{1}{s^2}e^{-sb}] - [0 - -\\frac{1}{s^2e^0}] \\\\\n&amp; b \\to \\infin \\Rightarrow [-\\frac{b}{s}e^{-sb} - \\frac{1}{s^2}e^{-sb}] \\to 0, \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\therefore &amp;\\mathcal{L}\\{t\\} = \\frac{1}{s^2}\n\\end{split}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\textcircled{3}\\;\\mathcal{L}\\{t^n\\} &amp; = \\int_0^\\infin {t^n}\\cdot{e^{-st}}dt \\\\\n\\text{let,}\\; &amp;u = t^n, dv = e^{-st}dt, u^\\prime = nt^{n-1}, v = -\\frac{1}{s}e^{-st}\\\\\n\\text{use,} &amp;\\int{udv} = uv - \\int u^\\prime v\\\\\n&amp;= \\lim\\limits_{b \\to \\infin}[-\\frac{t^n}{s}e^{-st}]_0^b - \\int_0^b{nt^{n-1}}(-\\frac{1}{s}e^{-st})dt \\\\\n&amp;\\dots \\\\\n&amp;= \\lim\\limits_{b \\to \\infin}[-\\frac{t^n}{s}e^{-st} - \\frac{nt^{n-1}}{s^2}e^{-st} - \\frac{n(n-1)t^{n-2}}{s^3}e^{-st}+ \\dots]_0^b \\\\\n\n\n\\end{split}<\/pre><\/div>\n\n\n\n<p>chain \ud615\ud0dc\uc758 \uc801\ubd84\uc740 \ub2e4\uc74c \ud14c\uc774\ube14\uc744 \uc774\uc6a9\ud574\uc11c \uc27d\uac8c \uac12\uc744 \uc5bb\uc744 \uc218 \uc788\ub2e4. <\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"489\" height=\"387\" src=\"http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/chain_int.png\" alt=\"\" class=\"wp-image-4589\" srcset=\"http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/chain_int.png 489w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/chain_int-300x237.png 300w\" sizes=\"auto, (max-width: 489px) 100vw, 489px\" \/><\/figure>\n<\/div>\n\n\n<p>\uccab\ubc88\uc9f8 \uceec\ub7fc\uc740 \ubbf8\ubd84\uc744 \ud574\ub098\uac00\uace0, \ub450\ubc88\uc9f8 \uceec\ub7fc\uc740 \uc801\ubd84\uc744 \ud574\ub098\uac00\ub294 \uac12\uc774\ub2e4. <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\text{when, }\\; &amp;b \\to \\infin , \\Rightarrow e^{sb} \\gg b^n\\\\\n&amp;=0 - (0 - 0 - \\dots + \\frac{n!}{s^{n+1}}) \\\\\n\\begin{aligned} \\end{aligned} \\\\\n&amp;\\therefore \\mathcal{L}\\{t^n \\} = \\frac{n!}{s^{n+1}}\n\\end{split}<\/pre><\/div>\n\n\n\n<p><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\textcircled{4}\\;\\mathcal{L}\\{e^{at}\\} &amp;=\\int_0^\\infin{e^{at}\\cdot e^{-st}dt} = \\int_0^\\infin{e^{-(s-a)t}dt} \\\\\n&amp;=\\lim\\limits_{b \\to \\infin} [ -\\frac{1}{s-a}e^{-(s-a)t} ]_0^b \\\\\n&amp;=\\lim\\limits_{b \\to \\infin}( -\\frac{1}{s-a}e^{-(s-a)b}) - (-\\frac{1}{s-a}e^0) \\\\\n&amp;\\text{if, }\\; s-a &lt;0 \\to \\text{diverge}, \\;s-a &gt; 0 \\to \\text{converges to 0,}\\\\ \n\\begin{aligned} \\end{aligned}\\\\\n\\therefore \\mathcal{L}\\{e^{at}\\}&amp;= \\frac{1}{s-a}, s&gt; a\n\\end{split}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\textcircled{5}\\;\\mathcal{L}\\{\\sin{kt}\\} = \\int _0^\\infin { e^{-st}\\cdot \\sin{kt}\\cdot dt } \\\\<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\text{let, } \\; u &amp;= e^{-st}, dv = \\sin{kt}\\cdot dt, du = -se^{-st}dt, v = -\\frac{1}{k}\\cos{kt} \\\\\n\\text{then,} \\;&amp;\\int{e^{-st}\\cdot \\sin{kt}\\cdot dt}= -\\frac{1}{k}e^{-st}\\cos{kt} - \\frac{s}{k}\\int e^{-st}\\cos{kt}dt \\\\\n\\text{once more let, } \\; u &amp;= e^{-st}, dv = \\cos{kt}dt, du = -se^{-st}dt, v = \\frac{1}{k}\\sin{kt} \\\\\n&amp;= -\\frac{1}{k}e^{-st}\\cos{kt} - \\frac{s}{k}[\\frac{1}{k}e^{-st}\\sin{kt} + \\frac{s}{k}\\int{e^{-st}\\sin{kt}dt}] \\\\\n&amp;=-\\frac{1}{k}e^{-st}\\cos{kt} - \\frac{1}{k^2}e^{-st}\\sin{kt} - \\frac{s^2}{k^2}\\int{e^{-st}\\sin{kt}dt} \\\\\n\\text{ \uc591\ubcc0\uc5d0 \ub9c8\uc9c0\ub9c9 } &amp;\\text{\uc801\ubd84\uc744 \uc0c1\uc1c4\ud558\ub294 \ud56d\uc744 \ub354\ud558\uba74 \uc801\ubd84\ud615 \uc0c1\uc1c4\ub418\uace0,}\\\\\n\\int{e^{-st}\\cdot \\sin{kt}\\cdot dt} +  &amp;\\frac{s^2}{k^2}\\int{e^{-st}\\sin{kt}dt}=-\\frac{1}{k}e^{-st}\\cos{kt} - \\frac{1}{k^2}e^{-st}\\sin{kt} \\\\\n\\frac{k^2+s^2}{k^2}\\int{e^{-st}\\cdot\\sin{kt}\\cdot dt} &amp;= -\\frac{1}{k}e^{-st}\\cos{kt} - \\frac{1}{k^2}e^{-st}\\sin{kt} \\\\\n\\begin{aligned} \\end{aligned}\\\\\n\\therefore \\int{e^{-st}\\cdot \\sin{kt}\\cdot dt} &amp;= \\frac{k^2}{k^2+s^2}(-\\frac{1}{k}e^{-st}\\cos{kt} - \\frac{1}{k^2}e^{-st}\\sin{kt})\n\\end{split}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\int _0^\\infin { e^{-st}\\cdot \\sin{kt}\\cdot dt } &amp;= \\lim\\limits_{b \\to \\infin} [\\frac{k^2}{k^2+s^2}(-\\frac{1}{k}e^{-st}\\cos{kt} - \\frac{1}{k^2}e^{-st}\\sin{kt})]_0^b\\\\\n&amp;0(b \\to \\infin) - \\frac{k^2}{k^2+s^2}(-\\frac{1}{k} - 0) \\\\\n\\begin{aligned} \\end{aligned}\\\\\n\\therefore \\mathcal{L}\\{\\sin{kt}\\} &amp;= \\frac{k}{k^2 + s^2}\n\\end{split}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\textcircled{6}\\;\\mathcal{L}\\{\\sinh{kt}\\} &amp;= \\frac{1}{2}\\int_0^\\infin{(e^{kt} - e^{-kt})e^{-st}dt} \\\\\n\\because \\sinh{kt} &amp;= \\frac{e^{kt} - e^{-kt}}{2} \\\\\n&amp;=\\frac{1}{2}\\int_0^\\infin{(e^{(k-s)t} - e^{-(k+s)t})}dt \\\\\n&amp;=\\lim\\limits_{b \\to \\infin}\\frac{1}{2}[\\frac{1}{k-s}e^{(k-s)t} - \\frac{1}{k+s}e^{-(k+s)t}]_0^b \\\\\n&amp;=\\frac{1}{2}[\\lim\\limits_{b \\to \\infin}(\\frac{1}{k-s}e^{(k-s)b}) + \\frac{1}{k+s}e^{-(k+s)b}) - \\frac{1}{k-s} + \\frac{1}{k+s}] \\\\\n&amp; \\text{only the convergence condition matters,}\\\\\n&amp;\\quad k&lt; s, (k&gt;0 , s &gt; 0 ) \\; or \\;(k &lt; 0, s&gt; 0) \\\\\n\\begin{aligned} \\end{aligned}\\\\\n\\therefore \\mathcal{L}\\{\\sinh{kt}\\} &amp;= \\frac{k}{s^2-k^2}\n\\end{split}<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"Linearity-of-Laplace-Transforms\">Linearity of Laplace Transforms<\/h2>\n\n\n\n<p>\ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658\uc758 \ud2b9\uc9d5\uc73c\ub85c \uc120\ud615\uc131\uc774 \uc788\ub2e4. <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\mathcal{L}\\{ {\\alpha f(t) + \\beta g(t)} \\} &amp;= \\int_0^\\infin{[\\alpha f(t) + \\beta g(t)]e^{-st}dt} \\\\\n&amp;= \\alpha \\int_0^\\infin{f(t)e^{-st}dt} + \\beta\\int_0^\\infin{g(t)e^{-st}dt} \\\\\n&amp;=\\alpha \\mathcal{L}\\{ f(t) \\} + \\beta \\mathcal{L}\\{ g(t) \\}\n\\end{split}<\/pre><\/div>\n\n\n\n<p>\uc55e\uc5d0\uc11c \uc720\ub3c4\ud588\ub358 \ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658\uc744 \ud14c\uc774\ube14\ub85c \uc815\ub9ac\ud574\ub450\uba74, \ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658 \ubc0f \uc5ed\ubcc0\ud658\uc744 \uc801\uc6a9\ud558\ub294\ub370 \ub3c4\uc6c0\uc774 \ub41c\ub2e4.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{array}{|c|c|}\n  \\hline\n  f(t) &amp; F(s)  \\\\\n  \\hline\n  1 &amp; 1\/s \\\\\n  \\hline\n  t^n &amp; n!\/s^{n+1} \\\\\n  \\hline\ne^at &amp; 1\/(s-a), (s&gt;a)\\\\\n  \\hline\n\\sin{kt} &amp; k\/(s^2+k^2)\\\\\n  \\hline\n\\cos{kt} &amp; s\/(s^2+k^2)\\\\\n\\hline\n\\sinh{kt} &amp; k\/(s^2-k^2)\\\\\n  \\hline\n\\cosh{kt} &amp; s\/(s^2-k^2)\\\\\n  \\hline\n\\end{array}<\/pre><\/div>\n\n\n\n<p>\uc120\ud615\uc131\uacfc \uc704 \ud45c\ub97c \uc0ac\uc6a9\ud558\uba74, \ubcf5\uc7a1\ud55c \ub2e4\ud56d\uc2dd\ub3c4 \uc190\uc27d\uac8c \ubcc0\ud658\/\uc5ed\ubcc0\ud658\uc774 \uac00\ub2a5\ud558\ub2e4.<\/p>\n\n\n\n<p>\uc608) laplace transform<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n&amp;\\mathcal{L}\\{ 3 - 5t + 2t^2 \\} \\\\\n&amp;=\\mathcal{L}\\{ 1 \\} -5 \\mathcal{L}\\{ t \\} +2\\mathcal{L}\\{ t^2 \\} \\\\\n&amp;= \\frac{3}{s} - 5\\frac{1}{s^2} + 2\\frac{2}{s^3}\n\\end{split}<\/pre><\/div>\n\n\n\n<p>\uc608) Inverse laplace transform<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\mathcal{L}^{-1}\\{ \\frac{5}{s}\\} &amp;= 5\\mathcal{L}^{-1}\\{ \\frac{1}{s}\\}  = 5 \\times1 = 5\\\\\n\\mathcal{L}^{-1}\\{ \\frac{4}{s^2}\\} &amp;=4\\mathcal{L}^{-1}\\{ \\frac{1}{s^{1+1}}\\} = 4t^1 = 4t\\\\\n\\mathcal{L}^{-1}\\{ \\frac{7}{s-3}\\} &amp;=7\\mathcal{L}^{-1}\\{ \\frac{1}{s-3}\\} = 7e^{3t}\\\\\n\\mathcal{L}^{-1}\\{ \\frac{7}{3s -1}\\} &amp;=\\frac{7}{3}\\mathcal{L}^{-1}\\{ \\frac{1}{s -1\/3}\\} = \\frac{7}{3}e^{-\\frac{1}{3}t}\\\\\n\\end{split}<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"Laplace-Transforms-of-Derivatives\">Laplace Transforms of Derivatives<\/h2>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\mathcal{L}\\{ f^\\prime(t)\\} &amp;= \\int_0^\\infin{e^{-st}f^\\prime (t)dt} \\\\\n\\text{let, } \\; &amp;u = e^{-st}, dv = f^\\prime(t)dt, du = -se^{-st}, v = f(t) \\\\\n&amp;= [e^{-st}f(t)]_0^\\infin - \\int_0^\\infin {(-se^{-st})f(t)dt} \\\\\nf(t) \\text{\uc758 \uc99d\uac00\uc728\uc774 } &amp;e^{-st} \\text{\ubcf4\ub2e4 \uc791\ub2e4\uace0 \uac00\uc815. \uc548\uadf8\ub7ec\uba74 diverse } \\\\\n&amp;= [0 - \\frac{f(0)}{e^0}] + s\\int_0^\\infin{e^{-st}f(t)dt} \\\\\n&amp;= s\\mathcal{L}\\{ f(t) \\} - f(0) \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\therefore \\mathcal{L}\\{ f^\\prime(t)\\} &amp;= sF(s) - f(0)\n\\end{split}<\/pre><\/div>\n\n\n\n<p>for higher order,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\mathcal{L}\\{ f^{\\prime\\prime}(t)\\} &amp;=\\mathcal{L}\\{ [f^\\prime(t)]^\\prime\\}\\\\\n&amp;= s\\mathcal{L}\\{ f^\\prime(t)\\} - f^\\prime(0) \\\\\n&amp;=s^2F(s) - sf(0) - f^\\prime(0) \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\mathcal{L}\\{ f^{\\prime\\prime\\prime}(t)\\} &amp;= s^3F(s) - s^2f(0) - sf^\\prime(0) - f^{\\prime\\prime}(0)\\\\\n&amp;\\dots \\\\\n\\mathcal{L}\\{ f^n(t) \\} &amp;= s^nF(s) - s^{(n-1)}f(0) - s^{(n-2)}f^\\prime(0) \\dots - f^{(n-1)}(0)\n\\end{split}<\/pre><\/div>\n\n\n\n<p>\uc608)<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\nf^{\\prime\\prime} - 2f^{\\prime} + f &amp;= 1 \\\\\n\\mathcal{L}\\{ f^{\\prime\\prime} - 2f^{\\prime} + f  \\} &amp;= \\mathcal{L}\\{1\\} \\\\\n[s^2F(s) - sf(0) - f^\\prime(0)] - 2[sF(s) - f(0)] + F(s) &amp;= \\frac{1}{s} \\\\\n(s^2-2s+1)F(s) - (s-2)f(0) - f^\\prime(0) &amp;= \\frac{1}{s} \\\\\nF(s) = \\frac{[\\frac{1}{s} + (s-2)f(0) + f^\\prime(0)]}{s^2 - 2s + 1}\n\\end{split}<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"Laplace-Transforms-to-Solve-D.E.\">Laplace Transforms to Solve D.E.<\/h2>\n\n\n\n<p>\uc608)<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y^{\\prime\\prime} - y = e^{2t}, \\quad y(0) = 0, y^\\prime(0) = 1<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\begin{cases}\n\\mathcal{L}\\{y \\} = F(s), \\\\\n\\mathcal{L}\\{y^\\prime \\} =sF(s) - y(0) = sF(s),  \\\\\n\\mathcal{L}\\{y^{\\prime\\prime} \\} =s^2F(s) - sy(0) - y^\\prime(0) = s^2F(s) - 1\n\\end{cases}\n\\end{split}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\mathcal{L}\\{ y^{\\prime\\prime} - y  \\} &amp;= [s^2F(s) - 1] - F(s) \\\\\n&amp;= (s^2-1)F(s) - 1 = \\mathcal{L}\\{ e^{2t}  \\} = \\frac{1}{s-2} \\\\\nF(s) &amp;= \\frac{s-1}{(s-2)(s^2-1)} = \\frac{s-1}{(s-2)(s-1)(s+1)} \\\\\n\\therefore F(s) &amp;= \\frac{1}{(s-2)(s+1)}\n\\end{split}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\nF(s) = \\frac{1}{(s-2)(s+1)} &amp;= \\frac{a}{s-2} + \\frac{b}{s+1}, \\quad \\text{find a, b}\\\\\n&amp;= \\frac{a(s+1) + b(s-2)}{(s-2)(s+1)} \\\\\n1 &amp;= a(s+1) + b(s-2) = (a+b)s + a-2b \\\\\na+b = 0, a-2b&amp;= 1 \\to a = -b, -3b = 1\\\\\n b &amp;= -\\frac{1}{3}, a =\\frac{1}{3} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\therefore F(s) &amp;= \\frac{1}{3}[\\frac{1}{s-2} - \\frac{1}{s+1}]\n\\end{split}<\/pre><\/div>\n\n\n\n<p>inverse laplace transform\uc744 \uc774\uc6a9,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\ny(t) &amp;= \\mathcal{L}^{-1}\\{ F(s) \\} = \\frac{1}{3}[\\mathcal{L}^{-1}\\{ \\frac{1}{s-2}\\}  - \\mathcal{L}^{-1}\\{ \\frac{1}{s+1}\\}] \\\\\n&amp;= \\frac{1}{3}(e^{2t} - e^{-t})\n\\end{split}<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"Unit-Step-Function\">Unit Step Function<\/h2>\n\n\n\n<p>\uc815\uc758<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>U(t-a) = \n\\begin{cases}\n0 &amp;0 \\le t \\lt a\\\\\n1 &amp; t \\ge a\n\\end{cases}<\/pre><\/div>\n\n\n\n<p>\uc544\ub798\ub294 a = 2\uc778 \uacbd\uc6b0\uc758 \uadf8\ub798\ud504 <\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" src=\"http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-unit-step01-1-300x300.png\" alt=\"\" class=\"wp-image-4675\" srcset=\"http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-unit-step01-1-300x300.png 300w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-unit-step01-1-150x150.png 150w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-unit-step01-1-768x768.png 768w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-unit-step01-1.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/figure>\n<\/div>\n\n\n<p>\uc774 unit step function\uc744 \ub2e4\ub978 \ud568\uc218\uc5d0 \uacf1\ud558\uba74, \ud2b9\uc815 \uad6c\uac04\uc744 switch off \uc2dc\ud0ac \uc218 \uc788\ub2e4.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\nf(t) = \\sqrt{t}, \\\\\ng(t) = \\sqrt{t}\\cdot U(t-2)\n\\end{split}<\/pre><\/div>\n\n\n\n<p>\uc704 \uc2dd\uc744 \uadf8\ub798\ud504\ub85c \uadf8\ub824\ubcf4\uba74,<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" src=\"http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-unit-step02-300x300.png\" alt=\"\" class=\"wp-image-4676\" srcset=\"http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-unit-step02-300x300.png 300w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-unit-step02-150x150.png 150w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-unit-step02-768x768.png 768w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-unit-step02.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/figure>\n<\/div>\n\n\n<p>U(t-2) \uac00 \ub179\uc0c9, f(t)\uac00 \ube68\uac15, g(t) \uac00 \ud30c\ub791 \uadf8\ub798\ud504\uc774\ub2e4. \uadf8\ub798\ud504\uc5d0\uc11c \ubcf4\uc774\ub4ef\uc774, f(t)\uc5d0 U(t-2)\ub97c \uacf1\ud558\uba74, 2\uae4c\uc9c0\ub294 \uadf8\ub798\ud504\uac00 0\uc774\uc5c8\ub2e4\uac00, 2\ubd80\ud130 \uc6d0\ub798 f(t)\uc758 \uadf8\ub798\ud504\uac00 \ud45c\uc2dc\ub418\ub294\uac78 \ubcfc \uc218 \uc788\ub2e4.<\/p>\n\n\n\n<p>time delay\ub3c4 \uac00\ub2a5\ud55c\ub370, U(t)\ub97c \uacf1\ud558\uba74\uc11c \uc6d0\ub798 \ud568\uc218\ub3c4 \uac19\uc774 shift \uc2dc\ucf1c\uc8fc\uba74, \ub2e4\uc74c\uacfc \uac19\uc774 \ub41c\ub2e4.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\nf(t) &amp;\\to f(t-2)\\cdot U(t-2) \\\\\nf(t)=\\cos\\left(t\\right) &amp;\\to  \\cos\\left(t-2\\right)u\\left(t-2\\right)\n\\end{split}<\/pre><\/div>\n\n\n\n<p>\uc704 \uc2dd\uacfc\uac19\uc774 \uc6d0\ub798 \ud568\uc218\uac00 cos(t)\uc77c \ub54c, u(t-2)\ub97c \uacf1\ud574\uc8fc\uba74\uc11c f(t) -&gt; t(t-2)\ub85c \uc774\ub3d9\uc2dc\ud0a4\uba74,<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" src=\"http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-unit-step03-1-300x300.png\" alt=\"\" class=\"wp-image-4680\" srcset=\"http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-unit-step03-1-300x300.png 300w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-unit-step03-1-150x150.png 150w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-unit-step03-1-768x768.png 768w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-unit-step03-1.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/figure>\n<\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"Laplace-Transforms-Involving-the-Unit-Step-Function\">Laplace Transforms Involving the Unit Step Function<\/h2>\n\n\n\n<p>Laplace transform\uacfc Unit Step function\uc744 \ub2e4\uc2dc \uc368\ubcf4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\mathcal{L}\\{ f(t) \\} &amp;= \\int_0^\\infin e^{-st}f(t)dt, \\\\\nU(t-a) &amp;= \\begin{cases}\n0 &amp;0 \\le t \\lt a\\\\\n1 &amp; t \\ge a\n\\end{cases}\n\\end{split}<\/pre><\/div>\n\n\n\n<p>\uc608)<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\mathcal{L}\\{ U(t-a) \\} &amp;= \\int_0^a{0\\cdot e^{-st}dt} + \\int_a^\\infin{1\\cdot e^{-st}dt} \\\\\n&amp;= [-\\frac{1}{s}e^{-st}]_a^\\infin \\\\\nt \\to \\infin &amp;\\Rightarrow e^{-st} \\to 0 \\quad \\text{\uc774\ubbc0\ub85c,}\\\\\n&amp;= 0 - (-\\frac{1}{s}e^{-sa}) \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\therefore \\mathcal{L}\\{ U(t-a) \\} &amp;= \\frac{e^{-sa}}{s}\n\\end{split}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\mathcal{L}\\{ f(t-a)U(t-a) \\} &amp;= \\int_0^a{0\\cdot U(t-a)e^{-st}dt}  + \\int_a^\\infin{1\\cdot f(t-a)e^{-st}dt}\\\\\n&amp;= \\int_0^\\infin{f(t-a)e^{-st}dt}\\\\\n\\text{let, } &amp;\\;w = t - a, dt = dw, t = w+a, \\\\\n&amp;=\\int_a^\\infin{e^{-s(w+a)}dw} = e^{-sa}\\int_a^\\infin{e^{-sw}f(w)dw} \\\\\n&amp;=e^{-sa}\\mathcal{L}\\{ f(w)  \\} = e^{-sa}\\mathcal{L}\\{ f(t-a) \\} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\therefore \\mathcal{L}\\{ f(t-a)U(t-a) \\} &amp;= e^{-as}F(s)\n\\end{split}<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"Impulse-and-Dirac-Delta-Function\">Impulse and Dirac Delta Function<\/h2>\n\n\n\n<p>\uc5ed\ud559\uc5d0\uc11c Impulse (\ucda9\uaca9\ub7c9)\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uac00\ud574\uc9c4 \ud798\uc744 \uc2dc\uac04\uc5d0 \ub300\ud574 \uc801\ubd84\ud55c \uac12\uc774\ub2e4. \uc774\uac83\uc740 \uc6b4\ub3d9\uc5d0\ub108\uc9c0\uc758 \uc801\ubd84\uc73c\ub85c\ubd80\ud130, \uc6b4\ub3d9\ub7c9\uc758 \ubcc0\ud654\ub7c9\uacfc \uac19\ub2e4\ub294\uac78 \uc54c \uc218 \uc788\ub2e4. F(t)\uac00 force\uc77c \ub54c,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\text{Impulse} &amp;= \\int_a^b{F(t)dt} \\\\\n&amp;=mv(b) - mv(a)\n\\end{split}<\/pre><\/div>\n\n\n\n<p>\ud798\uc774 \uac00\ud574\uc9c0\ub294 \uc2dc\uac04\uc774 \ub9e4\uc6b0 \uc9e7\uc544 0\uc73c\ub85c \uc218\ub834\ud558\ub294 \uacbd\uc6b0, t = 0\uc5d0\uc11c\ub9cc \ud568\uc218\uac12\uc774 \uc874\uc7ac\ud558\uace0 \uadf8 \uc678\uc5d0\ub294 0\uc774\ub418\ub294 \ud568\uc218\ub97c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub418\ub294 \ud568\uc218\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\frac{1}{k\\sqrt{\\pi}}\\cdot e^{-\\frac{t^2}{k^2}}<\/pre><\/div>\n\n\n\n<p>\uc774 \ud568\uc218\ub294 Gaussian-delta function\uc774\ub77c\uace0 \ubd88\ub9ac\uba70,  k\uac12\uc5d0 \ub530\ub77c \ud568\uc218\uc758 \ud3ed\uc774 \ubcc0\ud55c\ub2e4. <\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" src=\"http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-delta01-300x300.png\" alt=\"\" class=\"wp-image-4693\" srcset=\"http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-delta01-300x300.png 300w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-delta01-150x150.png 150w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-delta01-768x768.png 768w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-delta01.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/figure>\n<\/div>\n\n\n<p>\uc704 \uadf8\ub798\ud504\ub294 \uac80\uc815 : k=0.5, \ubcf4\ub77c: k = 0.1, \ub179\uc0c9: k = 0.01, \ube68\uac15: k = 0.0001\uc744 \uadf8\ub9b0 \uadf8\ub798\ud504\uc774\ub2e4. k\uac00 0\uc5d0 \uc218\ub834\ud560 \uc218\ub85d, \uadf8\ub798\ud504\ub294 0\uc5d0\uc11c\ub9cc \uac12\uc744 \uac16\ub294 \ub0a0\uce74\ub85c\uc6b4 \uadf8\ub798\ud504\uac00\ub418\uc9c0\ub9cc, \uc801\ubd84\uc744 \ud558\uba74 \uadf8 \uac12\uc774 1\uc774 \ub418\ub294 \ud568\uc218\ub2e4. <math data-latex=\"u = t\/(k\\sqrt{2})\"><semantics><mrow><mi>u<\/mi><mo>=<\/mo><mi>t<\/mi><mi>\/<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>k<\/mi><msqrt><mn>2<\/mn><\/msqrt><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">u = t\/(k\\sqrt{2})<\/annotation><\/semantics><\/math>  \ub85c \uce58\ud658\ud574\uc11c \uc801\ubd84\ud558\uba74 \uc27d\uac8c \ud655\uc778\uc774 \uac00\ub2a5\ud558\ub2e4. limit\ub97c \uc368\uc11c \ud45c\ud604\ud558\uba74, \ub2e4\uc74c\uacfc \uac19\uc774 \uc4f8 \uc218 \uc788\uc73c\uba70, \uc774\ub97c dirac-delta function\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\delta_k(t) = \\lim\\limits_{k \\to 0} \\frac{1}{k\\sqrt{\\pi}}\\cdot e^{-\\frac{t^2}{k^2}}<\/pre><\/div>\n\n\n\n<p>\uc0ac\uc2e4, \ud568\uc218\uc758 \ud615\ud0dc\ub294 \ud06c\uac8c \uc911\uc694\ud558\uc9c0 \uc54a\ub2e4. \uadf8\uc800, 0\uc73c\ub85c limit\ub97c \ubcf4\ub0bc \uc778\uc790\uc640 \uc801\ubd84\uc774 \ud56d\uc0c1 1\uc774\uba74 \ucda9\uc871\ud55c\ub2e4. \uc608\ub97c\ub4e4\uc5b4, <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y= \n\\begin{cases}\n\\frac{1}{2k}, &amp;\\quad(a-k\\lt x\\lt a+k) \\\\\n0, &amp;\\quad x \\lt a-k, x \\gt a+k\n\\end{cases}<\/pre><\/div>\n\n\n\n<p>\uc704\uc640\uac19\uc740 step function\uc744 \uc0dd\uac01\ud574\ubcf4\uba74, \uc801\ubd84\ud588\uc744 \ub54c 1\uc758 \uac12\uc744 \uac00\uc9c0\uba70(\uba74\uc801\uc774 \ud56d\uc0c1 1\uc774\ub2e4), \ub2e4\uc74c\uacfc \uac19\uc774 \uadf8\ub798\ud504\ub97c \uadf8\ub9b4 \uc218 \uc788\ub2e4. <\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" src=\"http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-delta03-300x300.png\" alt=\"\" class=\"wp-image-4697\" srcset=\"http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-delta03-300x300.png 300w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-delta03-150x150.png 150w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-delta03-768x768.png 768w, http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/04\/desmos-graph-delta03.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/figure>\n<\/div>\n\n\n<p>\uc774 \uadf8\ub798\ud504\ub294 <math data-latex=\"\\lim\\limits_{k \\to 0}\"><semantics><mrow><munder><mi>lim<\/mi><mrow><mi>k<\/mi><mo>\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><annotation encoding=\"application\/x-tex\">\\lim\\limits_{k \\to 0}<\/annotation><\/semantics><\/math> \ub97c \ucde8\ud588\uc744 \ub54c, \uc55e\uc5d0\uc11c \ub9d0\ud55c \uac83\ucc98\ub7fc 0\uc5d0\uc11c\ub9cc \uac12\uc744 \uac16\uc9c0\ub9cc \uc801\ubd84\ud588\uc744 \ub54c \uac12\uc774 1\uc774 \ub418\ub294 delta function \ud615\ud0dc\uac00 \ub41c\ub2e4. \uc774\ucc98\ub7fc, Dirac-delta function\uc740 \ud568\uc218\uc758 \ubaa8\uc591\ubcf4\ub2e4\ub294 \uadf8 \ud2b9\uc131\uc774 \uc911\uc694\ud558\ubbc0\ub85c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uace0 \uc0ac\uc6a9\uc774 \uac00\ub2a5\ud558\ub2e4.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\n\\text{Dirac-delta function :} &amp;\\quad \\delta{(t-t_0)} \\\\\n&amp;\\begin{cases}\n\\delta{(t-t_0)} = \\infin, &amp;t = t_0\\\\\n\\delta{(t-t_0)} = 0, &amp; t \\ne t_0\n\\end{cases}\\\\\n&amp;\\int_{-\\infin}^\\infin{\\delta(t-t_0)dt} = 1\n\\end{split}<\/pre><\/div>\n\n\n\n<p>\uc6d0\ub798, \uc774\ub7f0 \ud615\ud0dc\ub294 \uc218\ud559\uc5d0\uc11c \ub2e4\ub8f0 \uc218 \uc5c6\ub294 \ud2b9\uc774\uc810\uc5d0 \ud574\ub2f9\ud55c\ub2e4. \ud558\uc9c0\ub9cc, \uc55e\uc5d0\uc11c \ub9d0\ud55c \ucda9\uaca9\ub7c9\uc774\ub098 \uc810\uc804\ud558\uc640 \uac19\uc774 \ubb3c\ub9ac\uc5d0\uc11c \ub2e4\ub8e8\ub294 \uac12\ub4e4\uc774 \ud2b9\uc774\uc810\uc73c\ub85c \ubcf4\uc774\ub294 \ud615\ud0dc\ub77c\uc11c \uc774\ub4e4\uc744 \ub2e4\ub8e8\ub294\ub370 \uc774 Dirac-delta function\uc774 \uc0ac\uc6a9\ub418\uace0 \ub9e4\uc6b0 \uc720\uc6a9\ud558\ub2e4. <\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Laplacian transformation of Delta function<\/h3>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\mathcal{L}\\{ \\delta(t-t_0) \\} = \\int_0^\\infin{\\delta(t-t_0)e^{-st}dt} = 1\\cdot e^{-st} = e^{-st}<\/pre><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Solving D.E. with delta function<\/h3>\n\n\n\n<p>\uc608) <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y^\\prime + y = \\delta(t-1), \\quad y(0) = 2<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{cases}\n\\mathcal{L}\\{ y \\} &amp;= F(s) \\\\\n\\mathcal{L}\\{ y^\\prime \\} &amp;= sF(s)  - y(0) = sF(s) - 2 \\\\\n\\mathcal{L}\\{ \\delta(t-1) \\} &amp;= e^{-s} \\\\\n\\end{cases}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{split}\nsF(s) - 2 + F(s) &amp;= e^{-s} \\\\\n(s+1)F(s) &amp;= e^{-s} + 2 \\\\\n\\therefore F(s) &amp;= \\frac{e^{-s}+2}{s+ 1} \\\\\ny = \\mathcal{L}^{-1}\\{ \\frac{2}{s+1} \\} + \\mathcal{L}^{-1}\\{ \\frac{e^{-s}}{s+1} \\} &amp; = 2e^{-t} +  \\mathcal{L}^{-1}\\{ e^{-s}\\frac{{1}}{s+1} \\} \\\\\n\\text{from Unit step function,} \\quad &amp;\\mathcal{L}^{-1}\\{ e^{-sa}F(s) \\} = f(t-a)U(t-a), F(s) = \\frac{1}{s+1}\\\\\n&amp;=2e^{-t} + f(t-1)U(t-1)  = 2e^{-t} + e^{-(t-1)}U(t-1)\\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\therefore y = 2e^{-t} + e^{-(t-1)}U(t-1)\n\\end{split}<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"How-to-Shift-the-Index-for-Power-Series\">How to Shift the Index for Power Series<\/h2>\n\n\n\n<p>\uc6081)<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\Sigma_{n=1}^\\infin a_nx^{n-1} + \\Sigma_{n=0}^\\infin na_nx^{n}<\/pre><\/div>\n\n\n\n<p>\uc774\uac74 \ud558\ub098\uc758 \uc608\uc778\ub370, \uc704\uc640 \uac19\uc774 power series\uac00 \uc788\uc744 \ub54c, \uc774 \ub458\uc744 \ud569\uce58\ub294 \ubc29\ubc95\uc740 <\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li> x \uc758 \ucc28\uc218\ub97c \ub9de\ucd94\ub294 \ubc29\ubc95 <\/li>\n\n\n\n<li>n\uc758 \uc2dc\uc791 \uc778\ub371\uc2a4\ub97c \ub9de\ucd94\ub294 \ubc29\ubc95<\/li>\n<\/ol>\n\n\n\n<p>\ub450\uac00\uc9c0\uac00 \uc0ac\uc6a9\ub41c\ub2e4. \ubb34\ud55c\ub300\uc758 \uae09\uc218\uc774\uae30 \ub54c\ubb38\uc5d0, \uc774 \ub450\uac00\uc9c0\ub97c \ub9de\ucdb0\uc8fc\uba74 \uc55e\uc5d0 \uba87\uac1c \ud56d\uc774 \ub5a8\uc5b4\uc838 \ub098\uc624\uace0 \uadf8 \ub4a4\ub85c\ub294 \uac19\uc740 \ud615\ud0dc\uc758 \uc2dd\uc774 \ub41c\ub2e4. \uc704 \uc2dd\uc758 \uacbd\uc6b0,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\text{let, } \\; k = n-1, n = k+1 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\Sigma_{k=0}^\\infin a_{k+1}x^{k} + \\Sigma_{n=0}^\\infin na_nx^{n} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n= \\Sigma_{n=0}^\\infin (a_{n+1} +na_n)x^n<\/pre><\/div>\n\n\n\n<p>\uc6082)<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\Sigma_{n=0}^\\infin na_{n}x^{n-1} + \\Sigma_{n=0}^\\infin a_nx^{n+2} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\text{let, } \\; k = n-1, n = k+1\\quad \\text{and} \\quad j = n+2, n = j-2 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\Sigma_{k=0}^\\infin na_{k+1}x^{k} + \\Sigma_{j=2}^\\infin a_{j-2}x^{j} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n= a_1 + 2a_2x + \\Sigma_{n=2}^\\infin [(n+1)a_{n+1} + a_{n-2}]x^{n}<\/pre><\/div>\n\n\n\n<p>\uc6083)<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\Sigma_{n=2}^\\infin n(n-1)a_n x^{n-2} + x\\Sigma_{n=1}^\\infin na_n x^{n-1} + \\Sigma_{n=0}^\\infin a_n x^{n} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\text{let, } \\quad k = n-2, n = k+2 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\Sigma_{k=0}^\\infin {(k+1)(k+2)}a_{k+2} x^{k} + \\Sigma_{n=1}^\\infin {n}a_{n} x^{n} + \\Sigma_{n=0}^\\infin a_{n} x^{n} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n= a_0 + 2a_2 + \\Sigma_{n=1}^\\infin[(n+1)a_n + (n+1)(n+2)a_{n+2}]x^n<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"Solving-D.E.-with-Power-Series\">Solving D.E. with Power Series<\/h2>\n\n\n\n<p>\uc608) <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>y^{\\prime\\prime} - xy = 0 \\quad : \\text{Airy's Eq.}<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\text{let, } \\quad y = \\Sigma_{n=0}^\\infin a_nx^n \\\\\n\\begin{aligned} \\end{aligned} \\\\\ny\\prime = \\Sigma_{n=1}^\\infin na_nx^{n-1}  \\\\\n\\begin{aligned} \\end{aligned} \\\\\ny\\prime\\prime = \\Sigma_{n=2}^\\infin n(n-1)a_n x^{n-2}<\/pre><\/div>\n\n\n\n<p>\ubc29\uc815\uc2dd\uc5d0 \ub300\uc785\ud558\uba74,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\Sigma_{n=2}^\\infin n(n-1)a_n x^{n-2} - x \\Sigma_{n=0}^\\infin a_nx^n = 0 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\text{let, } \\quad k = n-2, n = k + 2, j = n+1, n = j-1\n\\begin{aligned} \\end{aligned} \\\\\n= \\Sigma_{k=0}^\\infin (k+1)(k+2)a_{k+2}x^k - \\Sigma_{j=1}^\\infin a_{j-1}x^{j} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n= 2a_2 + \\Sigma_{n=0}^\\infin[(n+1)(n+2)a_{n+2} - a_{n-1}]x^n = 0\\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\therefore a_2 = 0, (n+1)(n+2)a_{n+2} - a_{n-1} = 0 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\rightarrow a_{n+2} = \\frac{a_{k-1}}{a_{k+1}a_{k+2}}<\/pre><\/div>\n\n\n\n<p>\uc704\uc758 \uad00\uacc4\uc2dd\uc73c\ub85c\ubd80\ud130,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>a_3 = a_0\/5,  \\\\\na_4 = a_1\/12, \\\\\na_5 = a_2\/20 = 0, \\quad \\because a_2 = 0, \\\\\n\\vdots \\\\\n<\/pre><\/div>\n\n\n\n<p>3\ubc88\uc9f8 \ud56d\ub9c8\ub2e4 0\uc774\ub418\uace0, \ub098\uba38\uc9c0\ub294 <math data-latex=\"a_0, a_1\"><semantics><mrow><msub><mi>a<\/mi><mn>0<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>a<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">a_0, a_1<\/annotation><\/semantics><\/math> \uc5d0 \uc758\ud574 \uacb0\uc815\ub41c\ub2e4. \uc774 \ub450 \uac12\uc774 \uc790\uc720\ub3c4\ub97c \uac16\ub294\ub2e4\uace0 \ubcfc \uc218 \uc788\uace0, \uc11c\ub85c \ub3c5\ub9bd\uc801\uc778 \uc784\uc758\uc758 \uac12\uc744 \uac00\uc815\ud558\uba74, <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\text{let, } \\quad a_0 = 1, a_1 = 0 \\quad \\text{or} \\quad a_0 = 1, a_0 = 0\\\\\n\\begin{aligned} \\end{aligned} \\\\\ny_1(x) = a_0[1 + \\frac{x^3}{6} + \\dots] = 1 + \\frac{1}{6}x^3 + \\frac{1}{180}x^6 + \\dots \\\\\n\\begin{aligned} \\end{aligned} \\\\\ny_2(x) = a_1[x + \\frac{x^4}{12} + \\dots] = x + \\frac{1}{12}x^4 + \\frac{1}{504}x^7 + \\dots<\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"Solving-D.E-with-eigenvalue-and-eigenvector\">Solving D.E.s(Linear System) with eigenvalue and eigenvector<\/h2>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{aligned}\n\\frac{du}{dt} &amp;= \\lambda u \\\\\n\\text{solution :} \\quad u(t) &amp;= C\\cdot e^{\\lambda t} \\\\\n\\end{aligned} \\\\<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\text{for 2-D vector }\\;\\vec{u}, \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\frac{d\\vec{u}}{dt} = A\\vec{u} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\text{two equations have two solutions.} \\\\\n\\text{let, } \\; \\vec{x_1}\\; and\\; \\vec{x_2} \\; \\text{are solution vectors} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\text{generic solution:} \\quad u = C_1\\vec{x_1} + C_2\\vec{x_2} \n<\/pre><\/div>\n\n\n\n<p>\uc704\uc5d0\uc11c\ucc98\ub7fc,  <math data-latex=\"\\frac{d\\vec{u}}{dt} = A\\vec{u} \"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mover><mi>u<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><\/mrow><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mi>A<\/mi><mover><mi>u<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d\\vec{u}}{dt} = A\\vec{u} <\/annotation><\/semantics><\/math>    \ub85c \ud45c\ud604\ub418\ub294 n\uac1c\uc758 linear differential equation system\uc5d0\uc11c  <math data-latex=\"A\\vec{x} = \\lambda\\vec{x} \"><semantics><mrow><mi>A<\/mi><mover><mi>x<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>=<\/mo><mi>\u03bb<\/mi><mover><mi>x<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">A\\vec{x} = \\lambda\\vec{x} <\/annotation><\/semantics><\/math>   \ub97c \ub9cc\uc871\ud558\ub294 eigenvalue  <math data-latex=\"\\lambda\"><semantics><mi>\u03bb<\/mi><annotation encoding=\"application\/x-tex\">\\lambda<\/annotation><\/semantics><\/math>  , eignevector <math data-latex=\"\\vec{x}\"><semantics><mover><mi>x<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><annotation encoding=\"application\/x-tex\">\\vec{x}<\/annotation><\/semantics><\/math>  \ub97c \uad6c\ud558\uba74, <math data-latex=\"\\frac{d\\vec{u}}{dt} = A\\vec{u} \"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mover><mi>u<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><\/mrow><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mi>A<\/mi><mover><mi>u<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d\\vec{u}}{dt} = A\\vec{u} <\/annotation><\/semantics><\/math>  \ub97c  <math data-latex=\"\\frac{d\\vec{x}}{dt} = \\lambda \\vec{x}\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mover><mi>x<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><\/mrow><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mi>\u03bb<\/mi><mover><mi>x<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d\\vec{x}}{dt} = \\lambda \\vec{x}<\/annotation><\/semantics><\/math>  \uc640 \uac19\uc774 \uc4f8 \uc218 \uc788\uace0, \uadf8 \ud574\ub294 <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\vec{u(t)} = \\Sigma_{i=1}^n {C_i}\\cdot\\vec{x_i}\\cdot e^{\\lambda_i t}<\/pre><\/div>\n\n\n\n<p>\uc880 \ub354 \uad6c\uccb4\uc801\uc73c\ub85c 2 by 2 matrix\uc758 \uacbd\uc6b0\ub97c \uc0dd\uac01\ud574\ubcf4\uba74,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{bmatrix}\n\\frac{dx_1}{dt} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\frac{dx_2}{dt} \\\\\n\\end{bmatrix} =\n\\begin{bmatrix}\na &amp; b \\\\\n\\begin{aligned} \\end{aligned} \\\\\nc &amp; d \\\\\n\\end{bmatrix} \n\\begin{bmatrix}\nx_1 \\\\\n\\begin{aligned} \\end{aligned} \\\\\nx_2 \\\\\n\\end{bmatrix}\n<\/pre><\/div>\n\n\n\n<p><math data-latex=\"A\\vec{u} = \\lambda \\vec{u}\"><semantics><mrow><mi>A<\/mi><mover><mi>u<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>=<\/mo><mi>\u03bb<\/mi><mover><mi>u<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">A\\vec{u} = \\lambda \\vec{u}<\/annotation><\/semantics><\/math>  \ub97c \uba3c\uc800 \ud47c\ub2e4. <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{bmatrix}\na &amp; b \\\\\n\\begin{aligned} \\end{aligned} \\\\\nc &amp; d \\\\\n\\end{bmatrix} \n\\begin{bmatrix}\nu_{11} \\\\\n\\begin{aligned} \\end{aligned} \\\\\nu_{12} \\\\\n\\end{bmatrix} = \n\\lambda\n\\begin{bmatrix}\nu_{11} \\\\\n\\begin{aligned} \\end{aligned} \\\\\nu_{12} \\\\\n\\end{bmatrix}<\/pre><\/div>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-2e912ea9d1eacd8bfbd33d327c9d23a9\" style=\"color:#b03232\"><math data-latex=\"u_{11}\"><semantics><msub><mi>u<\/mi><mn>11<\/mn><\/msub><annotation encoding=\"application\/x-tex\">u_{11}<\/annotation><\/semantics><\/math> \uac19\uc740 \ud45c\uae30\ub294 \ub9e4\ud2b8\ub9ad\uc2a4 \uc5d8\ub9ac\uba3c\ud2b8\uc640 \ud63c\ub3d9\uc774 \uc788\uc9c0\ub9cc, \uc5ec\uae30\uc11c\ub294 \ubca1\ud130 <math data-latex=\"\\vec{u_1}\"><semantics><mover><msub><mi>u<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><annotation encoding=\"application\/x-tex\">\\vec{u_1}<\/annotation><\/semantics><\/math>  \uc758 \ub450 \uc131\ubd84\ud45c\uc2dc\ub97c \uc774\ub807\uac8c \ud558\uaca0\ub2e4. <\/p>\n\n\n\n<p>\uc774\ub85c\ubd80\ud130 eigenvalue <math data-latex=\"\\lambda_1, \\lambda_2\"><semantics><mrow><msub><mi>\u03bb<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>\u03bb<\/mi><mn>2<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\lambda_1, \\lambda_2<\/annotation><\/semantics><\/math> \uc640 eigenvector <math data-latex=\"\\vec{u_1}, \\vec{u_2}\"><semantics><mrow><mover><msub><mi>u<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo separator=\"true\">,<\/mo><mover><msub><mi>u<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\vec{u_1}, \\vec{u_2}<\/annotation><\/semantics><\/math> \ub97c \uad6c\ud588\ub2e4\uace0 \ud558\uba74,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{bmatrix}\n\\frac{dx_1}{dt} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\frac{dx_2}{dt} \\\\\n\\end{bmatrix} =\n\\lambda\n\\begin{bmatrix}\nu_{11} \\\\\n\\begin{aligned} \\end{aligned} \\\\\nu_{12} \\\\\n\\end{bmatrix} \\\\\n\\begin{aligned} \\end{aligned} \\\\\nx_1 =  C_1\\cdot\\vec{u_1}\\cdot e^{\\lambda_1 t}, x_2 = C_2\\cdot \\vec{u_2}\\cdot e^{\\lambda_2 t} \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\text{generic solution : }\\quad \\vec{x} = C_1\\cdot\\vec{u_1}\\cdot e^{\\lambda_1 t} +  C_2\\cdot \\vec{u_2}\\cdot e^{\\lambda_2 t} <\/pre><\/div>\n\n\n\n<p>\uc608) <\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{cases}\n\\frac{dx}{dt} = 4x - y \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\frac{dy}{dt} = 2x + y \\\\\n\\end{cases}\n\\quad \\rightarrow \\quad \nA = \n\\begin{bmatrix}\n4 &amp; -1 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n2 &amp; 1 \\\\\n\\end{bmatrix} \n,\\quad\ndet(A-\\lambda I) = 0\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>det(A - \\lambda I) = \n\\begin{vmatrix}\n4-\\lambda &amp; -1 \\\\\n2 &amp; 1-\\lambda\n\\end{vmatrix} = (4-\\lambda)(1-\\lambda) + 2 = \\lambda^2 -5\\lambda+6 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n(\\lambda-2)(\\lambda-3) = 0, \\lambda = 2, 3<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{aligned}\n\\text{i)} \\lambda_1 = 2 : &amp;\\\\\n&amp; (A-2I)\\vec{u} = 0,\\quad \\vec{u} \\;\\text{is eigenvector}\\\\\n&amp;\\begin{bmatrix}\n4-2 &amp; -1 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n2 &amp; 1-2 \\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nu_{11} \\\\\n\\begin{aligned} \\end{aligned} \\\\\nu_{12} \\\\\n\\end{bmatrix}\n =\n\\begin{bmatrix}\n2 &amp; -1 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n2 &amp; -1 \\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nu_{11} \\\\\n\\begin{aligned} \\end{aligned} \\\\\nu_{12} \\\\\n\\end{bmatrix}\n\\end{aligned} \\\\\n2u_{11} = u_{12} \\\\\n\\text{let, }\\; u_{11} = 1 \\;\\text{then,}\\; u_{12} = 2 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\vec{u_1} = \n\\begin{bmatrix}\n1 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n2 \\\\\n\\end{bmatrix}\n<\/pre><\/div>\n\n\n\n<p>\ub9c8\ucc2c\uac00\uc9c0\ub85c,<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\begin{aligned}\n\\text{ii)} \\lambda_2 = 3: &amp; \\\\\n&amp;\\begin{bmatrix}\n4-3 &amp; -1 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n2 &amp; 1-3 \\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nu_{21} \\\\\n\\begin{aligned} \\end{aligned} \\\\\nu_{22} \\\\\n\\end{bmatrix}\n =\n&amp;\\begin{bmatrix}\n1 &amp; -1 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n2 &amp; -2 \\\\\n\\end{bmatrix}\n\\end{aligned} \\\\\n\n\n\\begin{bmatrix}\nu_{21} \\\\\n\\begin{aligned} \\end{aligned} \\\\\nu_{22} \\\\\n\\end{bmatrix} \\\\\n\\begin{aligned} \\end{aligned} \\\\\nu_{21} = u_{22}, \\quad \\text{let, } \\; u_{21} = 1 \\; \\text{then, } \\; u_{22} = 1 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n\\vec{u_2} = \\begin{bmatrix}\n1 \\\\\n\\begin{aligned} \\end{aligned} \\\\\n1 \\\\\n\\end{bmatrix} \\\\<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\therefore \\vec{x} = \nC_1\n\\begin{bmatrix}\n1 \\\\\n2\n\\end{bmatrix}\ne^{2t} + \nC_1\n\\begin{bmatrix}\n1 \\\\\n1\n\\end{bmatrix}\ne^{3t}<\/pre><\/div>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Directly Integrable D.E. \uadf8\ub0e5 \uc801\ubd84\ud558\uba74 \ub418\ub294 \ud615\ud0dc. Separable D.E. \uc67c\ucabd\uc740 y\uc5d0 \ub300\ud574\uc11c\ub9cc, \uc624\ub978\ucabd x\uc5d0 \ub300\ud574\uc11c\ub9cc \ub098\uc624\ub3c4\ub85d \uc815\ub9ac. \uc591\ucabd\uc744 \uac01\uac01 y, x\uc5d0 \ub300\ud574\uc11c \uc801\ubd84\ud558\uba74 \ub428. \uc6081) \uc6082) Linear 1st. order eq. : Intergrating factor Integrating factor \ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4. \uc704\uc5d0\uc11c \uad6c\ud55c integrating factor\ub97c \uc6d0\ub798 \uc2dd\uc758 \uc591\ubcc0\uc5d0 \uacf1\ud558\uba74, \uacb0\uacfc\uc801\uc73c\ub85c integrating factor\ub97c \uacf1\ud574\uc90c\uc73c\ub85c <a href=\"http:\/\/batmask.net\/index.php\/differential-equation\/\" class=\"btn btn-link continue-link\">\ub354 \uc77d\uae30<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-4238","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"http:\/\/batmask.net\/index.php\/wp-json\/wp\/v2\/pages\/4238","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/batmask.net\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/batmask.net\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/batmask.net\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/batmask.net\/index.php\/wp-json\/wp\/v2\/comments?post=4238"}],"version-history":[{"count":429,"href":"http:\/\/batmask.net\/index.php\/wp-json\/wp\/v2\/pages\/4238\/revisions"}],"predecessor-version":[{"id":4715,"href":"http:\/\/batmask.net\/index.php\/wp-json\/wp\/v2\/pages\/4238\/revisions\/4715"}],"wp:attachment":[{"href":"http:\/\/batmask.net\/index.php\/wp-json\/wp\/v2\/media?parent=4238"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}