Internal problem ID [2288]
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 24.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+5 y^{\prime }+4 y-F \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 47
dsolve(diff(y(x),x$2)+5*diff(y(x),x)+4*y(x)=F(x),y(x), singsol=all)
\[ y \relax (x ) = c_{2} {\mathrm e}^{-x}+{\mathrm e}^{-4 x} c_{1}+\frac {\left (\left (\int {\mathrm e}^{x} F \relax (x )d x \right ) {\mathrm e}^{3 x}-\left (\int F \relax (x ) {\mathrm e}^{4 x}d x \right )\right ) {\mathrm e}^{-4 x}}{3} \]
✓ Solution by Mathematica
Time used: 0.035 (sec). Leaf size: 61
DSolve[y''[x]+5*y'[x]+4*y[x]==F[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-4 x} \left (\int _1^x-\frac {1}{3} e^{4 K[1]} F(K[1])dK[1]+e^{3 x} \left (\int _1^x\frac {1}{3} e^{K[2]} F(K[2])dK[2]+c_2\right )+c_1\right ) \\ \end{align*}