73.3.2 problem 7 (ii)

Internal problem ID [19812]
Book : Elementary Differential Equations. By R.L.E. Schwarzenberger. Chapman and Hall. London. First Edition (1969)
Section : Chapter 5. Linear equations. Exercises at page 85
Problem number : 7 (ii)
Date solved : Thursday, October 02, 2025 at 04:44:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

With initial conditions

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 14

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

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

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

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

\begin{align*} x(t)&\to \frac {1}{4} e^{-t} \left (e^{2 t} (2 t+1)-1\right ) \end{align*}

✓ Sympy. Time used: 0.066 (sec). Leaf size: 19

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

\[ x{\left (t \right )} = \left (\frac {t}{2} + \frac {1}{4}\right ) e^{t} - \frac {e^{- t}}{4} \]