85.59.8 problem 8

Internal problem ID [22849]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. A Exercises at page 203
Problem number : 8
Date solved : Thursday, October 02, 2025 at 09:15:44 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 30

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

\[ y = {\mathrm e}^{x} c_2 +{\mathrm e}^{-x} c_1 +\frac {x \left (2 x^{2}-3 x +3\right ) {\mathrm e}^{x}}{12} \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 40

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

\begin{align*} y(x)&\to \frac {1}{24} e^x \left (4 x^3-6 x^2+6 x-3+24 c_1\right )+c_2 e^{-x} \end{align*}

✓ Sympy. Time used: 0.084 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = C_{2} e^{- x} + \left (C_{1} + \frac {x^{3}}{6} - \frac {x^{2}}{4} + \frac {x}{4}\right ) e^{x} \]