26.9.7 problem Exercise 22.7, page 240

Internal problem ID [7116]
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 22.7, page 240
Date solved : Tuesday, September 30, 2025 at 04:21:38 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 19

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

\[ y = {\mathrm e}^{-x} \left (c_2 +c_1 x +\frac {1}{12} x^{4}\right ) \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 27

ode=D[y[x],{x,2}]+2*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 {1}{12} e^{-x} \left (x^4+12 c_2 x+12 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.162 (sec). Leaf size: 15

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

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + \frac {x^{3}}{12}\right )\right ) e^{- x} \]