78.2.42 problem 17.b

Internal problem ID [20994]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 2, Second order ODEs. Problems section 2.6
Problem number : 17.b
Date solved : Thursday, October 02, 2025 at 07:01:13 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-y&={\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: 0.028 (sec). Leaf size: 16

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

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

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 27

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

✓ Sympy. Time used: 0.060 (sec). Leaf size: 20

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

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