12.10.13 problem 13

Internal problem ID [1817]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.7 Variation of Parameters. Page 262
Problem number : 13
Date solved : Monday, January 27, 2025 at 05:36:02 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} \left (2 x +1\right ) y^{\prime \prime }-2 y^{\prime }-\left (2 x +3\right ) y&=\left (2 x +1\right )^{2} {\mathrm e}^{-x} \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 28

dsolve((2*x+1)*diff(y(x),x$2)-2*diff(y(x),x)-(2*x+3)*y(x)=(2*x+1)^2*exp(-x),y(x), singsol=all)
 

\[ y = \frac {\left (-x^{2}+2 c_2 -x \right ) {\mathrm e}^{-x}}{2}+x \,{\mathrm e}^{x} c_1 \]

✓ Solution by Mathematica

Time used: 0.077 (sec). Leaf size: 42

DSolve[(2*x+1)*D[y[x],{x,2}]-2*D[y[x],x]-(2*x+3)*y[x]==(2*x+1)^2*Exp[-x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\frac {1}{2} e^{-x} x (x+1)+c_1 e^{-x-\frac {1}{2}}+c_2 e^{x+\frac {1}{2}} x \]