20.9.24 problem Problem 24

Internal problem ID [3768]
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
Date solved : Monday, January 27, 2025 at 08:01:06 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+5 y^{\prime }+4 y&=F \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 45

dsolve(diff(y(x),x$2)+5*diff(y(x),x)+4*y(x)=F(x),y(x), singsol=all)
 

\[ y \left (x \right ) = c_{2} {\mathrm e}^{-x}+c_{1} {\mathrm e}^{-4 x}+\frac {\left (\int {\mathrm e}^{x} F \left (x \right )d x \right ) {\mathrm e}^{-x}}{3}-\frac {{\mathrm e}^{-4 x} \left (\int F \left (x \right ) {\mathrm e}^{4 x}d x \right )}{3} \]

✓ Solution by Mathematica

Time used: 0.058 (sec). Leaf size: 66

DSolve[D[y[x],{x,2}]+5*D[y[x],x]+4*y[x]==F[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-4 x} \left (\int _1^x-\frac {1}{3} e^{4 K[1]} F(K[1])dK[1]+e^{3 x} \int _1^x\frac {1}{3} e^{K[2]} F(K[2])dK[2]+c_2 e^{3 x}+c_1\right ) \]