20.9.25 problem Problem 25

Internal problem ID [3769]
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 25
Date solved : Monday, January 27, 2025 at 08:01:07 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }-2 y&=F \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 46

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

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

✓ Solution by Mathematica

Time used: 0.052 (sec). Leaf size: 68

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

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