44.11.20 problem 5(e)

Internal problem ID [9290]
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 : Tuesday, September 30, 2025 at 06:16:11 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*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 25

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

\[ y = x \left (\operatorname {Ei}_{1}\left (x \right ) x +c_1 x +\operatorname {Ei}_{1}\left (x \right )-{\mathrm e}^{-x}+c_2 \right ) \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 53

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

\begin{align*} y(x)&\to x \left (\int _1^x-\frac {e^{-K[1]}}{K[1]}dK[1]+x \int _1^x\frac {e^{-K[2]}}{K[2]^2}dK[2]+c_2 x+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.551 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - 2*x*Derivative(y(x), x) - x*exp(-x) + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x \left (C_{1} + C_{2} x - x \operatorname {Ei}{\left (x e^{i \pi } \right )} - \operatorname {Ei}{\left (x e^{i \pi } \right )} - e^{- x}\right ) \]