9.4 Variation of Parameters for Higher Order Equations

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}\pgfmathsetmacro {\rabrot }{\sinphi }\pgfmathsetmacro {\racrot }{0}\pgfmathsetmacro {\rbarot }{-\ctsp }\pgfmathsetmacro {\rbbrot }{\ctcp }\pgfmathsetmacro {\rbcrot }{\sintheta }\pgfmathsetmacro {\rcarot }{\stsp }\pgfmathsetmacro {\rcbrot }{-\stcp }\pgfmathsetmacro {\rccrot }{\costheta }} \newcommand {\tdplotcalctransformrotmain }[0]{\tdplotsinandcos {\sinalpha }{\cosalpha }{\tdplotalpha } \tdplotsinandcos {\sinbeta }{\cosbeta }{\tdplotbeta } \tdplotsinandcos {\singamma }{\cosgamma }{\tdplotgamma } \tdplotmult {\sasb }{\sinalpha }{\sinbeta } \tdplotmult {\sbsg }{\sinbeta }{\singamma } \tdplotmult {\sasg }{\sinalpha }{\singamma } \tdplotmult {\sasbsg }{\sasb }{\singamma } \tdplotmult {\sacb }{\sinalpha }{\cosbeta } \tdplotmult {\sacg }{\sinalpha }{\cosgamma } \tdplotmult {\sbcg }{\sinbeta }{\cosgamma } \tdplotmult {\sacbsg }{\sacb }{\singamma } \tdplotmult {\sacbcg }{\sacb }{\cosgamma } \tdplotmult {\casb }{\cosalpha }{\sinbeta } \tdplotmult {\cacb }{\cosalpha }{\cosbeta } \tdplotmult 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}{\cosgamma } \tdplotmult {\sbcg }{\sinbeta }{\cosgamma } \tdplotmult {\sacbsg }{\sacb }{\singamma } \tdplotmult {\sacbcg }{\sacb }{\cosgamma } \tdplotmult {\casb }{\cosalpha }{\sinbeta } \tdplotmult {\cacb }{\cosalpha }{\cosbeta } \tdplotmult {\cacg }{\cosalpha }{\cosgamma } \tdplotmult {\casg }{\cosalpha }{\singamma } \tdplotmult {\cacbsg }{\cacb }{\singamma } \tdplotmult {\cacbcg }{\cacb }{\cosgamma } \pgfmathsetmacro {\raaeul }{\cacbcg -\sasg } \pgfmathsetmacro {\rabeul }{\sacbcg + \casg } \pgfmathsetmacro {\raceul }{-\sbcg } \pgfmathsetmacro {\rbaeul }{-\cacbsg -\sacg } \pgfmathsetmacro {\rbbeul }{-\sacbsg + \cacg } \pgfmathsetmacro {\rbceul }{\sbsg } \pgfmathsetmacro {\rcaeul }{\casb } \pgfmathsetmacro {\rcbeul }{\sasb } \pgfmathsetmacro {\rcceul }{\cosbeta } } \newcommand {\tdplottransformmainrot }[3]{\tdplotcalctransformmainrot \par \pgfmathsetmacro {\tdplotresx }{\raaeul * ##1 + \rabeul * ##2 + \raceul * ##3} \pgfmathsetmacro {\tdplotresy }{\rbaeul * ##1 + \rbbeul * ##2 + \rbceul * ##3} \pgfmathsetmacro {\tdplotresz }{\rcaeul * ##1 + \rcbeul * ##2 + \rcceul * ##3} } \newcommand {\tdplottransformrotmain }[3]{\tdplotcalctransformrotmain \par \pgfmathsetmacro {\tdplotresx }{\raaeul * ##1 + \rabeul * ##2 + \raceul * ##3} \pgfmathsetmacro {\tdplotresy }{\rbaeul * ##1 + \rbbeul * ##2 + \rbceul * ##3} \pgfmathsetmacro {\tdplotresz }{\rcaeul * ##1 + \rcbeul * ##2 + \rcceul * ##3} } \newcommand {\tdplottransformmainscreen }[3]{\tdplotcalctransformmainscreen \par \pgfmathsetmacro {\tdplotresx }{\raarot * ##1 + \rabrot * ##2 + \racrot * ##3} \pgfmathsetmacro {\tdplotresy }{\rbarot * ##1 + \rbbrot * ##2 + \rbcrot * ##3} } \newcommand {\tdplotsetrotatedcoords }[3]{\pgfmathsetmacro {\tdplotalpha }{##1} \pgfmathsetmacro {\tdplotbeta }{##2} \pgfmathsetmacro {\tdplotgamma }{##3} \tdplotcalctransformrotmain \par \tdplotmult {\raaeaa }{\raarot }{\raaeul } \tdplotmult {\rabeba }{\rabrot }{\rbaeul } \tdplotmult {\raceca }{\racrot }{\rcaeul } \tdplotmult {\raaeab }{\raarot }{\rabeul } \tdplotmult {\rabebb }{\rabrot }{\rbbeul } \tdplotmult {\racecb }{\racrot }{\rcbeul } \tdplotmult {\raaeac }{\raarot }{\raceul } \tdplotmult {\rabebc }{\rabrot }{\rbceul } \tdplotmult {\racecc }{\racrot }{\rcceul } \tdplotmult {\rbaeaa }{\rbarot }{\raaeul } \tdplotmult {\rbbeba }{\rbbrot }{\rbaeul } \tdplotmult {\rbceca }{\rbcrot }{\rcaeul } \tdplotmult {\rbaeab }{\rbarot }{\rabeul } \tdplotmult {\rbbebb }{\rbbrot }{\rbbeul } \tdplotmult {\rbcecb }{\rbcrot }{\rcbeul } \tdplotmult {\rbaeac }{\rbarot }{\raceul } \tdplotmult {\rbbebc }{\rbbrot }{\rbceul } \tdplotmult {\rbcecc }{\rbcrot }{\rcceul } \pgfmathsetmacro {\raarc }{\raaeaa + \rabeba + \raceca } \pgfmathsetmacro {\rabrc }{\raaeab + \rabebb + \racecb } \pgfmathsetmacro {\racrc }{\raaeac + \rabebc + \racecc } \pgfmathsetmacro {\rbarc }{\rbaeaa + \rbbeba + \rbceca } \pgfmathsetmacro {\rbbrc }{\rbaeab + \rbbebb + \rbcecb } \pgfmathsetmacro {\rbcrc }{\rbaeac + \rbbebc + \rbcecc } \tikzset {tdplot_rotated_coords/.append style={x={(\raarc cm,\rbarc cm)},y={(\rabrc cm,\rbbrc cm)},z={(\racrc cm,\rbcrc cm)}}}} \newcommand {\tdplotsetrotatedcoordsorigin }[1]{\tikzset {tdplot_rotated_coords/.append style={shift=##1}}} \newcommand {\tdplotresetrotatedcoordsorigin }[0]{\tikzset {tdplot_rotated_coords/.append style={shift={(0,0,0)}}}} \newcommand {\tdplotsetthetaplanecoords }[1]{\tdplotresetrotatedcoordsorigin \tdplotsetrotatedcoords {270 + ##1}{270}{0}} \newcommand {\tdplotsetrotatedthetaplanecoords }[1]{\tdplotsetrotatedcoords {\tdplotalpha }{\tdplotbeta }{\tdplotgamma + ##1}\tikzset {tdplot_rotated_coords/.append style={y={(\raarc cm,\rbarc cm)},z={(\rabrc cm,\rbbrc cm)},x={(\racrc cm,\rbcrc cm)}}}} \newcommand {\tdplotsetcoord }[4]{\tdplotsinandcos {\sinthetavec }{\costhetavec }{##3}\tdplotsinandcos {\sinphivec }{\cosphivec }{##4}\tdplotmult {\stcpv }{\sinthetavec }{\cosphivec }\tdplotmult {\stspv }{\sinthetavec }{\sinphivec }\coordinate (##1) at ($##2*(\stcpv ,\stspv ,\costhetavec )$); \coordinate (##1xy) at ($##2*(\stcpv ,\stspv ,0)$); \coordinate (##1xz) at ($##2*(\stcpv ,0,\costhetavec )$); \coordinate (##1yz) at ($##2*(0,\stspv ,\costhetavec )$); \coordinate (##1x) at ($##2*(\stcpv ,0,0)$); \coordinate (##1y) at ($##2*(0,\stspv ,0)$); \coordinate (##1z) at ($##2*(0,0,\costhetavec )$); } \newcommand {\tdplotsimplesetcoord }[4]{\tdplotsinandcos {\sinthetavec }{\costhetavec }{##3}\tdplotsinandcos {\sinphivec }{\cosphivec }{##4}\tdplotmult {\stcpv }{\sinthetavec }{\cosphivec }\tdplotmult {\stspv }{\sinthetavec }{\sinphivec }\coordinate (##1) at ($##2*(\stcpv ,\stspv ,\costhetavec )$); } \newcommand {\tdplotsetpolarplotrange }[4]{\pgfmathsetmacro {\tdplotlowerphi }{##3} \pgfmathsetmacro {\tdplotupperphi }{##4} \pgfmathsetmacro {\tdplotlowertheta }{##1} \pgfmathsetmacro {\tdplotuppertheta }{##2} } \newcommand {\tdplotresetpolarplotrange }[0]{\pgfmathsetmacro {\tdplotlowerphi }{0} \pgfmathsetmacro {\tdplotupperphi }{360} \pgfmathsetmacro {\tdplotlowertheta }{0} \pgfmathsetmacro {\tdplotuppertheta }{180} } \newcommand {\tdplotdosurfaceplot }[6]{\par \pgfmathsetmacro {\nextphi }{\curphi + \tdplotsuperfudge *\viewphistep } \par \begin {scope}[opacity=1] \par \par \tdplotcheckdiff {\nextphi }{360}{\origviewphistep }{##2}{} \tdplotcheckdiff {\nextphi }{0}{\origviewphistep }{##2}{} \par \tdplotcheckdiff {\nextphi }{90}{\origviewphistep }{##3}{} \tdplotcheckdiff {\nextphi }{450}{\origviewphistep }{##3}{} \end {scope} \par \foreach \curtheta in{\viewthetastart ,\viewthetainc ,...,\viewthetaend } { \par \pgfmathsetmacro {\curlongitude }{90 -\curphi } \pgfmathsetmacro {\curlatitude }{90 -\curtheta } \par \ifthenelse {\equal {\leftright }{-1.0}}{\pgfmathsetmacro {\curphi }{\curphi -\origviewphistep } }{} \par \pgfmathsetmacro {\tdplottheta }{mod(\curtheta ,360)} \pgfmathsetmacro {\tdplotphi }{mod(\curphi ,360)} \par \pgfmathparse {\tdplotphi <0} \ifthenelse {\equal {\pgfmathresult }{1}}{ \pgfmathsetmacro {\tdplotphi }{\tdplotphi + 360} }{}\par \pgfmathparse {\tdplottheta >\tdplotuppertheta } \pgfmathsetmacro {\logictest }{1 -\pgfmathresult } \par \pgfmathparse {\tdplottheta <\tdplotlowertheta } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \pgfmathsetmacro {\tdplottheta }{\tdplottheta + \viewthetastep } \pgfmathparse {\tdplottheta >\tdplotuppertheta } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \pgfmathparse {\tdplottheta <\tdplotlowertheta } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \pgfmathparse {\tdplotphi >\tdplotupperphi } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \pgfmathparse {\tdplotphi <\tdplotlowerphi } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \pgfmathsetmacro {\tdplotphi }{\tdplotphi + \viewphistep } \par \pgfmathparse {\tdplotphi <0} \ifthenelse {\equal {\pgfmathresult }{1}}{ \pgfmathsetmacro {\tdplotphi }{\tdplotphi + 360} }{}\par \pgfmathparse {\tdplotphi >\tdplotupperphi } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \pgfmathparse {\tdplotphi <\tdplotlowerphi } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \par \pgfmathsetmacro {\tdplottheta }{\curtheta } \pgfmathsetmacro {\tdplotphi }{\curphi } \par \ifthenelse {\equal {##6}{parametricfill}}{\ifthenelse {\equal {\logictest }{1.0}}{\pgfmathsetmacro {\radius }{##1} \pgfmathsetmacro {\tdplotr }{\radius *360} \par \pgfmathlessthan {\radius }{0} \pgfmathsetmacro {\phaseshift }{180 * \pgfmathresult } \par \pgfmathsetmacro {\colorarg }{##5} \pgfmathsetmacro {\colorarg }{\colorarg + \phaseshift } \pgfmathsetmacro {\colorarg }{mod(\colorarg ,360)} \par \pgfmathlessthan {\colorarg }{0} \pgfmathsetmacro {\colorarg }{\colorarg + 360*\pgfmathresult } \par \pgfmathdivide {\colorarg }{360} \definecolor {tdplotfillcolor}{hsb}{\pgfmathresult ,1,1} \color {tdplotfillcolor} }{}}{\pgfsetfillcolor {##5} } \pgfsetstrokecolor {##4} \par \ifthenelse {\equal {\leftright 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{\tdplothuestep }{5} \pgfmathsetmacro {\tdplotxsize }{##3} \pgfmathsetmacro {\tdplotysize }{##4} \par \pgfmathsetmacro {\tdplotyscale }{\tdplotysize /360} \par \foreach \tdplotphi in {0,\tdplothuestep ,...,360} { \pgfmathdivide {\tdplotphi }{360} \definecolor {tdplotfillcolor}{hsb}{\pgfmathresult ,1,1} \color {tdplotfillcolor} \par \pgfmathsetmacro {\tdplotstarty }{\tdploty + \tdplotphi * \tdplotyscale } \pgfmathsetmacro {\tdplotstopy }{\tdplotstarty + \tdplothuestep * \tdplotyscale } \pgfmathsetmacro {\tdplotstartx }{\tdplotx } \pgfmathsetmacro {\tdplotstopx }{\tdplotx + \tdplotxsize } \filldraw [tdplot_screen_coords] (\tdplotstartx ,\tdplotstarty ) rectangle (\tdplotstopx ,\tdplotstopy ); } \par \pgfmathsetmacro {\tdplotstopy }{\tdploty + (360+\tdplothuestep )*\tdplotyscale } \pgfmathsetmacro {\tdplotstopx }{\tdplotx + \tdplotxsize } \par \draw [tdplot_screen_coords] (\tdplotx ,\tdploty ) rectangle (\tdplotstopx ,\tdplotstopy ); \par \node 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We discuss the solution of an $n$ th order nonhomogeneous linear differential equation, making use of variation of parameters to find a particular solution.

Variation of Parameters for Higher Order Equations

Derivation of the method

We assume throughout this section that the nonhomogeneous linear equation

$\begin{equation} \label{eq:9.4.1} P_0(x)y^{(n)}+P_1(x)y^{(n-1)}+\cdots+P_n(x)y=F(x) \end{equation}$ is normal on an interval $(a,b)$ . We’ll abbreviate this equation as $Ly=F$ , where $Ly=P_0(x)y^{(n)}+P_1(x)y^{(n-1)}+\cdots +P_n(x)y.$ When we speak of solutions of this equation and its complementary equation $Ly=0$ , we mean solutions on $(a,b)$ . We’ll show how to use the method of variation of parameters to find a particular solution of $Ly=F$ , provided that we know a fundamental set of solutions $\{y_1,y_2,\dots ,y_n\}$ of $Ly=0$ .

We seek a particular solution of $Ly=F$ in the form

$\begin{equation} \label{eq:9.4.2} y_p=u_1y_1+u_2y_2+\cdots+u_ny_n \end{equation}$ where $\{y_1,y_2,\dots ,y_n\}$ is a known fundamental set of solutions of the complementary equation $P_0(x)y^{(n)}+P_1(x)y^{(n-1)}+\cdots +P_n(x)y=0$ and $u_1, u_2, \dots , u_n$ are functions to be determined. We begin by imposing the following $n-1$ conditions on $u_1,u_2,\dots ,u_n$ : $\begin{equation} \label{eq:9.4.3} \begin{array}{rcl} u'_1y_1+u'_2y_2+&\cdots&+u'_ny_n=0 \\ u'_1y'_1+u'_2y'_2+&\cdots&+u'_ny'_n=0 \\ &\vdots& \\ u'_1y_1^{(n-2)}+u'_2y^{(n-2)}_2+&\cdots&+u'_ny^{(n-2)}_n =0. \end{array} \end{equation}$ These conditions lead to simple formulas for the first $n-1$ derivatives of $y_p$ : $\begin{equation} \label{eq:9.4.4} y^{(r)}_p=u_1y^{(r)}_1+u_2y_2^{(r)}\cdots+u_ny^{(r)}_n,\ 0 \leq r \leq n-1. \end{equation}$ These formulas are easy to remember, since they look as though we obtained them by differentiating (eq:9.4.2) $n-1$ times while treating $u_1, u_2, \dots , u_n$ as constants. To see that (eq:9.4.3) implies (eq:9.4.4), we first differentiate (eq:9.4.2) to obtain $y_p'=u_1y_1'+u_2y_2'+\cdots +u_ny_n'+u_1'y_1+u_2'y_2+\cdots +u_n'y_n,$ which reduces to $y_p'=u_1y_1'+u_2y_2'+\cdots +u_ny_n'$ because of the first equation in (eq:9.4.3). Differentiating this yields $y_p''=u_1y_1''+u_2y_2''+\cdots +u_ny_n''+u_1'y_1'+u_2'y_2'+\cdots +u_n'y_n',$ which reduces to $y_p''=u_1y_1''+u_2y_2''+\cdots +u_ny_n''$ because of the second equation in (eq:9.4.3). Continuing in this way yields (eq:9.4.4).

The last equation in (eq:9.4.4) is $y_p^{(n-1)}=u_1y_1^{(n-1)}+u_2y_2^{(n-1)}+\cdots +u_ny_n^{(n-1)}.$ Differentiating this yields $y_p^{(n)}=u_1y_1^{(n)}+u_2y_2^{(n)}+\cdots +u_ny_n^{(n)}+ u_1'y_1^{(n-1)}+u_2'y_2^{(n-1)}+\cdots +u_n'y_n^{(n-1)}.$ Substituting this and (eq:9.4.4) into (eq:9.4.1) yields $u_1Ly_1+u_2Ly_2+\cdots +u_nLy_n+P_0(x)\left ( u_1'y_1^{(n-1)}+u_2'y_2^{(n-1)}+\cdots +u_n'y_n^{(n-1)}\right )=F(x).$ Since $Ly_i=0$ $(1 \leq i \leq n)$ , this reduces to $u_1'y_1^{(n-1)}+u_2'y_2^{(n-1)}+\cdots +u_n'y_n^{(n-1)}=\frac {F(x)}{P_0(x)}.$ Combining this equation with (eq:9.4.3) shows that $y_p=u_1y_1+u_2y_2+\cdots +u_ny_n$ is a solution of (eq:9.4.1) if $\begin {array}{rcl} u'_1y_1+u'_2y_2+&\cdots &+u'_ny_n=0 \\ u'_1y'_1+u'_2y'_2+&\cdots &+u'_ny'_n=0 \\ &\vdots & \\ u'_1y_1^{(n-2)}+u'_2y^{(n-2)}_2+&\cdots &+u'_ny^{(n-2)}_n =0 \\ u'_1y^{(n-1)}_1+u'_2y^{(n-1)}_2+&\cdots &+u'_n y^{(n-1)}_n=F/P_0, \end {array}$ which can be written in matrix form as

$\begin{equation} \label{eq:9.4.5} \begin{bmatrix} y_1&y_2&\cdots&y_n \\ y'_1&y'_2&\cdots&y_n'\\ \vdots&\vdots&\ddots&\vdots\\ y_1^{(n-2)}&y_2^{(n-2)}&\cdots&y_n^{(n-2)}\\ y_1^{(n-1)}&y_2^{(n-1)}&\cdots&y_n^{(n-1)} \end{bmatrix} \begin{bmatrix}u_1'\\u_2'\\\vdots\\u_{n-1}'\\u_n'\end{bmatrix}= \begin{bmatrix}0\\0\\ \vdots\\0\\F/ P_0\end{bmatrix}. \end{equation}$ The determinant of this system is the Wronskian $W$ of the fundamental set of solutions $\{y_1,y_2,\dots ,y_n\}$ , which has no zeros on $(a,b)$ , by Theorem thmtype:9.1.4. Solving (eq:9.4.5) by Cramer’s rule yields $\begin{equation} \label{eq:9.4.6} u'_j=(-1)^{n-j}\frac{FW_j}{P_0W},\quad 1\leq j\leq n, \end{equation}$ where $W_j$ is the Wronskian of the set of functions obtained by deleting $y_j$ from $\{y_1,y_2,\dots ,y_n\}$ and keeping the remaining functions in the same order. Equivalently, $W_j$ is the determinant obtained by deleting the last row and $j$ -th column of $W$ .

Having obtained $u_1', u_2', \dots , u_n'$ , we can integrate to obtain $u_1,\,u_2,\dots ,u_n$ . As in Trench 5.7, we take the constants of integration to be zero, and we drop any linear combination of $\{y_1,y_2,\dots ,y_n\}$ that may appear in $y_p$ .

For efficiency, it’s best to compute $W_1, W_2, \dots , W_n$ first, and then compute $W$ by expanding in cofactors of the last row; thus, $W=\sum _{j=1}^n(-1)^{n-j}y_j^{(n-1)}W_j.$

Third Order Equations

If $n=3$ , then $W=\begin {vmatrix} y_1&y_2&y_3 \\ y'_1&y'_2&y'_3 \\ y''_1&y''_2&y''_3 \end {vmatrix}$ Therefore $W_1=\begin {vmatrix} y_2&y_3 \\ y'_2&y'_3 \end {vmatrix}, \quad W_2=\begin {vmatrix} y_1&y_3 \\ y'_1&y'_3 \end {vmatrix}, \quad W_3=\begin {vmatrix} y_1&y_2 \\ y'_1&y'_2 \end {vmatrix},$ and (eq:9.4.6) becomes

$\begin{equation} \label{eq:9.4.7} u'_1=\frac{FW_1}{P_0W},\quad u'_2=-\frac{FW_2}{P_0W},\quad u'_3=\frac{FW_3}{P_0W}. \end{equation}$

Find a particular solution of $\begin{equation} \label{eq:9.4.8} xy'''-y''-xy'+y=8x^2e^x, \end{equation}$ given that $y_1=x$ , $y_2=e^x$ , and $y_3=e^{-x}$ form a fundamental set of solutions of the complementary equation. Then find the general solution of (eq:9.4.8).

We seek a particular solution of (eq:9.4.8) of the form $y_p=u_1x+u_2e^x+u_3e^{-x}.$ The Wronskian of $\{y_1,y_2,y_3\}$ is $W(x)=\begin {vmatrix} x&e^x&e^{-x} \\ 1&e^x&-e^{-x} \\ 0&e^x&e^{-x} \end {vmatrix},$ so

$\begin{eqnarray*} W_1&=& \begin{vmatrix} e^x&e^{-x}\\ e^x&-e^{-x} \end{vmatrix}=-2,\\ W_2&=& \begin{vmatrix} x&e^{-x}\\1&-e^{-x} \end{vmatrix}=-e^{-x}(x+1),\\ W_3&=& \begin{vmatrix} x&e^x\\1&e^x \end{vmatrix}=e^x(x-1). \end{eqnarray*}$

Expanding $W$ by cofactors of the last row yields $W=0W_1-e^x W_2+e^{-x}W_3=0(-2)-e^x\left (-e^{-x}(x+1)\right ) +e^{-x}e^x(x-1)=2x.$ Since $F(x)=8x^2e^x$ and $P_0(x)=x$ , $\frac {F}{P_0W}=\frac {8x^2e^x}{x\cdot 2x}=4e^x.$ Therefore, from (eq:9.4.7)

$\begin{eqnarray*} u'_1&=& 4e^xW_1=4e^x(-2)=-8e^x,\\ u_2'&=&-4e^xW_2=-4e^x\left(-e^{-x}(x+1)\right)=4(x+1),\\ u_3'&=&4e^xW_3=4e^x\left(e^x(x-1)\right)=4e^{2x}(x-1). \end{eqnarray*}$

Integrating and taking the constants of integration to be zero yields $u_1=-8e^x,\quad u_2=2(x+1)^2, u_3=e^{2x}(2x-3).$ Hence,

$\begin{eqnarray*} y_p&=&u_1y_1+u_2y_2+u_3y_3\\ &=&(-8e^x)x+e^x(2(x+1)^2)+e^{-x}\left(e^{2x}(2x-3)\right) \\&=&e^x(2x^2-2x-1). \end{eqnarray*}$

Since $-e^x$ is a solution of the complementary equation, we redefine $y_p=2xe^x(x-1).$ Therefore the general solution of (eq:9.4.8) is $y=2xe^x(x-1)+c_1x+c_2e^x+c_3e^{-x}.$

Fourth Order Equations

If $n=4$ , then $W=\begin {vmatrix} y_1&y_2&y_3&y_4 \\ y'_1&y'_2&y'_3&y_4' \\ y''_1&y''_2&y''_3&y_4''\\ y'''_1&y'''_2&y'''_3&y_4''' \end {vmatrix},$ Therefore $W_1=\begin {vmatrix} y_2&y_3&y_4 \\ y'_2&y'_3&y_4'\\ y''_2&y''_3&y_4'' \end {vmatrix}, \quad W_2=\begin {vmatrix} y_1&y_3&y_4 \\ y'_1&y'_3&y_4'\\ y''_1&y''_3&y_4'' \end {vmatrix},$

$W_3=\begin {vmatrix} y_1&y_2&y_4 \\ y'_1&y'_2&y_4'\\ y''_1&y''_2&y_4'' \end {vmatrix},\quad W_4=\begin {vmatrix} y_1&y_2&y_3 \\ y_1'&y'_2&y_3'\\ y_1''&y''_2&y_3'' \end {vmatrix},$ and (eq:9.4.6) becomes

$\begin{equation} \label{eq:9.4.9} u'_1=-\frac{FW_1}{P_0W},\quad u'_2=\frac{FW_2}{P_0W},\quad u'_3=-\frac{FW_3}{P_0W},\quad u'_4=\frac{FW_4}{P_0W}. \end{equation}$

Find a particular solution of $\begin{equation} \label{eq:9.4.10} x^4y^{(4)}+6x^3y'''+2x^2y''-4xy'+4y=12x^2, \end{equation}$ given that $y_1=x$ , $y_2=x^2$ , $y_3=1/x$ and $y_4=1/x^2$ form a fundamental set of solutions of the complementary equation. Then find the general solution of (eq:9.4.10) on $(-\infty ,0)$ and $(0,\infty )$ .

We seek a particular solution of (eq:9.4.10) of the form $y_p=u_1x+u_2x^2+\frac {u_3}{x}+\frac {u_4}{x^2}.$ The Wronskian of $\{y_1,y_2,y_3,y_4\}$ is $W(x)=\begin {vmatrix} x&x^2&1/x&-1/x^2 \\ 1&2x&-1/x^2&-2/x^3 \\ 0&2&2/x^3&6/x^4\\ 0&0&-6/x^4&-24/x^5 \end {vmatrix},$ so

$\begin{eqnarray*} W_1&=& \begin{vmatrix} x^2&1/x&1/x^2\\ 2x&-1/x^2&-2/x^3\\ 2&2/x^3&6/x^4 \end{vmatrix}=-\frac{12}{x^4},\\ W_2&=& \begin{vmatrix} x&1/x&1/x^2\\ 1&-1/x^2&-2/x^3\\ 0&2/x^3&6/x^4 \end{vmatrix}=-\frac{6}{x^5},\\ W_3&=& \begin{vmatrix} x&x^2&1/x^2\\ 1&2x&-2/x^3\\ 0&2&6/x^4 \end{vmatrix} =\frac{12}{x^2}, \\ W_4&=& \begin{vmatrix} x&x^2&1/x\\ 1&2x&-1/x^2\\ 0&2&2/x^3 \end{vmatrix}=\frac{6}{x}. \end{eqnarray*}$

Expanding $W$ by cofactors of the last row yields

$\begin{eqnarray*} W&=&-0W_1+0 W_2-\left(-\frac{6}{x^4}\right)W_3+\left(-\frac{24}{x^5}\right)W_4\\ &=&\frac{6}{x^4}\frac{12}{x^2}-\frac{24}{x^5}\frac{6}{x}=-\frac{72}{x^6}. \end{eqnarray*}$

Since $F(x)=12x^2$ and $P_0(x)=x^4$ , $\frac {F}{P_0W}=\frac {12x^2}{x^4}\left (-\frac {x^6}{72}\right )=-\frac {x^4}{6}.$ Therefore, from (eq:9.4.9),

$\begin{eqnarray*} u'_1&=&-\left(-\frac{x^4}{6}\right)W_1=\frac{x^4}{6}\left(-\frac{12}{x^4}\right)=-2,\\ u_2'&=&-\frac{x^4}{6}W_2=-\frac{x^4}{6}\left(-\frac{6}{x^5}\right) =\frac{1}{x},\\ u_3'&=&-\left(-\frac{x^4}{6}\right)W_3=\frac{x^4}{6}\frac{12}{x^2}=2x^2,\\ u_4'&=&-\frac{x^4}{6}W_4=-\frac{x^4}{6}\frac{6}{x}=-x^3 \end{eqnarray*}$

Integrating these and taking the constants of integration to be zero yields $u_1=-2x,\quad u_2=\ln |x|,\quad u_3=\frac {2x^3}{3}, u_4=-\frac {x^4}{4}.$ Hence,

$\begin{eqnarray*} y_p&=&u_1y_1+u_2y_2+u_3y_3+u_4y_4\\ &=&(-2x)x+(\ln|x|)x^2+\frac{2x^3}{3}\frac{1}{x}+\left(-\frac{x^4}{4}\right) \frac{1}{x^2} \\&=&x^2\ln|x|-\frac{19x^2}{12}. \end{eqnarray*}$

Since $-19x^2/12$ is a solution of the complementary equation, we redefine $y_p=x^2\ln |x|.$ Therefore $y=x^2\ln |x|+c_1x+c_2x^2+\frac {c_3}{x}+\frac {c_4}{x^2}$ is the general solution of (eq:9.4.10) on $(-\infty ,0)$ and $(0,\infty )$ .

Text Source

Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)

https://digitalcommons.trinity.edu/mono/8/