{"id":4238,"date":"2026-04-05T15:28:44","date_gmt":"2026-04-05T06:28:44","guid":{"rendered":"http:\/\/batmask.net\/?page_id=4238"},"modified":"2026-04-05T22:37:54","modified_gmt":"2026-04-05T13:37:54","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>Laplace Transform<\/li>\n\n\n\n<li>Linearity of Laplace Transforms<\/li>\n\n\n\n<li>Laplace Transforms of Derivatives<\/li>\n\n\n\n<li>Laplace Transforms to Solve D.E.<\/li>\n\n\n\n<li>Unit Step Function<\/li>\n\n\n\n<li>Laplace Transforms Involving the Unit Step Function<\/li>\n\n\n\n<li>Impulse and Dirac Delta Function<\/li>\n\n\n\n<li>How to shift the Index for Power Series<\/li>\n\n\n\n<li>Solving D.E. with Power Series<\/li>\n\n\n\n<li>Solving D.E. Linear system with eigenvalue and eigenvector<\/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)dy = 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-full\"><img loading=\"lazy\" decoding=\"async\" width=\"995\" height=\"600\" src=\"http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/03\/diff_eq01.png\" alt=\"\" class=\"wp-image-4271\" srcset=\"http:\/\/batmask.net\/wordpress\/wp-content\/uploads\/2026\/03\/diff_eq01.png 995w, 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\" sizes=\"auto, (max-width: 995px) 100vw, 995px\" \/><\/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<p>    <\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"cauchy-euler-eq\">Cauchy-Euler Eq.<\/h2>\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 \ud574\uc11c, seperable \ud615\ud0dc\ub85c \ubc14\ub01c. \uc608 [&hellip;]<\/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":182,"href":"http:\/\/batmask.net\/index.php\/wp-json\/wp\/v2\/pages\/4238\/revisions"}],"predecessor-version":[{"id":4438,"href":"http:\/\/batmask.net\/index.php\/wp-json\/wp\/v2\/pages\/4238\/revisions\/4438"}],"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}]}}