50.11.20 problem 5(e)

Internal problem ID [8004]
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)
Date solved : Monday, January 27, 2025 at 03:36:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=x \,{\mathrm e}^{-x} \end{align*}

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 25

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

\[ y = x \left (\operatorname {Ei}_{1}\left (x \right ) x +c_{2} x +\operatorname {Ei}_{1}\left (x \right )-{\mathrm e}^{-x}+c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.049 (sec). Leaf size: 30

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

\[ y(x)\to x \left (-(x+1) \operatorname {ExpIntegralEi}(-x)-e^{-x}+c_2 x+c_1\right ) \]