87.17.33 problem 34

Internal problem ID [23625]
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 : 34
Date solved : Thursday, October 02, 2025 at 09:43:36 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 20

ode:=diff(diff(y(x),x),x)-y(x) = x*exp(x); 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\left (x^{2}-x \right ) {\mathrm e}^{x}}{4}+\frac {5 \sinh \left (x \right )}{4} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 32

ode=D[y[x],{x,2}]-y[x]==x*Exp[x]; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{8} e^{-x} \left (e^{2 x} \left (2 x^2-2 x+5\right )-5\right ) \end{align*}

✓ Sympy. Time used: 0.089 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*exp(x) - y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {x^{2}}{4} - \frac {x}{4} + \frac {5}{8}\right ) e^{x} - \frac {5 e^{- x}}{8} \]