26.9.19 problem Exercise 22, problem 19, page 240

Internal problem ID [7128]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 4. Higher order linear differential equations. Lesson 22. Variation of Parameters
Problem number : Exercise 22, problem 19, page 240
Date solved : Tuesday, September 30, 2025 at 04:21:48 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 24

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

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

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 51

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

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

✓ Sympy. Time used: 0.194 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = \frac {C_{1}}{x} + C_{2} x + e^{- x} + \frac {e^{- x}}{x} \]