Internal problem ID [5582]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 2. Second-Order Linear Equations. Section 2.3. THE METHOD OF VARIATION OF
PARAMETERS. Page 71
Problem number: 5(d).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y-{\mathrm e}^{2 x} x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 23
dsolve(x*diff(y(x),x$2)-(1+x)*diff(y(x),x)+y(x)=x^2*exp(2*x),y(x), singsol=all)
\[ y \relax (x ) = \left (x +1\right ) c_{2}+c_{1} {\mathrm e}^{x}+\frac {\left (x -1\right ) {\mathrm e}^{2 x}}{2} \]
✓ Solution by Mathematica
Time used: 0.045 (sec). Leaf size: 31
DSolve[x*y''[x]-(1+x)*y'[x]+y[x]==x^2*Exp[2*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} e^{2 x} (x-1)+c_1 e^x-c_2 (x+1) \\ \end{align*}