20.9.26 problem Problem 26

Internal problem ID [3770]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.7, The Variation of Parameters Method. page 556
Problem number : Problem 26
Date solved : Monday, January 27, 2025 at 08:01:08 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }-12 y&=F \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 45

dsolve(diff(y(x),x$2)+4*diff(y(x),x)-12*y(x)=F(x),y(x), singsol=all)
 

\[ y \left (x \right ) = -\frac {\left (-\left (\int F \left (x \right ) {\mathrm e}^{-2 x}d x \right ) {\mathrm e}^{8 x}-8 c_{1} {\mathrm e}^{8 x}+\int F \left (x \right ) {\mathrm e}^{6 x}d x -8 c_{2} \right ) {\mathrm e}^{-6 x}}{8} \]

✓ Solution by Mathematica

Time used: 0.069 (sec). Leaf size: 68

DSolve[D[y[x],{x,2}]+4*D[y[x],x]-12*y[x]==F[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-6 x} \left (\int _1^x-\frac {1}{8} e^{6 K[1]} F(K[1])dK[1]+e^{8 x} \int _1^x\frac {1}{8} e^{-2 K[2]} F(K[2])dK[2]+c_2 e^{8 x}+c_1\right ) \]