Internal problem ID [5583]
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(e).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y-{\mathrm e}^{-x} x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 29
dsolve(x^2*diff(y(x),x$2)-2*x*diff(y(x),x)+2*y(x)=x*exp(-x),y(x), singsol=all)
\[ y \relax (x ) = x^{2} c_{2}+c_{1} x +\left (\left (x +1\right ) \expIntegral \left (1, x\right )-{\mathrm e}^{-x}\right ) x \]
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 29
DSolve[x^2*y''[x]-2*x*y'[x]+2*y[x]==x*Exp[-x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x (-(x+1) \text {Ei}(-x)+\sinh (x)-\cosh (x)+c_2 x+c_1) \\ \end{align*}