87.17.23 problem 23

Internal problem ID [23615]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 127
Problem number : 23
Date solved : Thursday, October 02, 2025 at 09:43:29 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 39

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

\[ y = -\frac {3}{4}+\frac {\left (-9 x^{2}+27 c_2 -6 x -2\right ) {\mathrm e}^{-x}}{27}-\frac {x^{2}}{2}+{\mathrm e}^{2 x} c_1 +\frac {x}{2} \]

✓ Mathematica. Time used: 0.138 (sec). Leaf size: 51

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

\begin{align*} y(x)&\to \frac {1}{4} \left (-2 x^2+2 x-3\right )-\frac {1}{27} e^{-x} \left (9 x^2+6 x+2-27 c_1\right )+c_2 e^{2 x} \end{align*}

✓ Sympy. Time used: 0.184 (sec). Leaf size: 36

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

\[ y{\left (x \right )} = C_{2} e^{2 x} - \frac {x^{2}}{2} + \frac {x}{2} + \left (C_{1} - \frac {x^{2}}{3} - \frac {2 x}{9}\right ) e^{- x} - \frac {3}{4} \]