73.3.1 problem 7 (i)

Internal problem ID [19811]
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 (i)
Date solved : Thursday, October 02, 2025 at 04:44:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{\prime \prime }-x&=t^{2} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 21

ode:=diff(diff(x(t),t),t)-x(t) = t^2; 
ic:=[x(0) = 0, D(x)(0) = 1]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {3 \,{\mathrm e}^{t}}{2}+\frac {{\mathrm e}^{-t}}{2}-t^{2}-2 \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 27

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

\begin{align*} x(t)&\to \frac {1}{2} \left (-2 \left (t^2+2\right )+e^{-t}+3 e^t\right ) \end{align*}

✓ Sympy. Time used: 0.051 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-t**2 - x(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)
 

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