Internal problem ID [2797]
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]]
\[ \boxed {y^{\prime \prime }+5 y^{\prime }+4 y=F \left (x \right )} \]
✓ 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)
\[ y \left (x \right ) = {\mathrm e}^{-x} c_{2} +c_{1} {\mathrm e}^{-4 x}+\frac {\left (\left (\int {\mathrm e}^{x} F \left (x \right )d x \right ) {\mathrm e}^{3 x}-\left (\int F \left (x \right ) {\mathrm e}^{4 x}d x \right )\right ) {\mathrm e}^{-4 x}}{3} \]
✓ Solution by Mathematica
Time used: 0.06 (sec). Leaf size: 66
DSolve[y''[x]+5*y'[x]+4*y[x]==F[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-4 x} \left (\int _1^x-\frac {1}{3} e^{4 K[1]} F(K[1])dK[1]+e^{3 x} \int _1^x\frac {1}{3} e^{K[2]} F(K[2])dK[2]+c_2 e^{3 x}+c_1\right ) \]