59.7.2 problem 14.1 (ii)

Internal problem ID [15047]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 14, Inhomogeneous second order linear equations. Exercises page 140
Problem number : 14.1 (ii)
Date solved : Thursday, October 02, 2025 at 10:02:22 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} x^{\prime \prime }-4 x^{\prime }&=t^{2} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 26

ode:=diff(diff(x(t),t),t)-4*diff(x(t),t) = t^2; 
dsolve(ode,x(t), singsol=all);
 

\[ x = -\frac {t^{2}}{16}-\frac {t^{3}}{12}+\frac {{\mathrm e}^{4 t} c_1}{4}-\frac {t}{32}+c_2 \]

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 36

ode=D[x[t],{t,2}]-4*D[x[t],t]==t^2; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{96} \left (-8 t^3-6 t^2-3 t+24 c_1 e^{4 t}+96 c_2\right ) \end{align*}

✓ Sympy. Time used: 0.098 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-t**2 - 4*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = C_{1} + C_{2} e^{4 t} - \frac {t^{3}}{12} - \frac {t^{2}}{16} - \frac {t}{32} \]