86.11.10 problem 10 (a)

Internal problem ID [23204]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 9. The operational method. Exercise 9c at page 137
Problem number : 10 (a)
Date solved : Thursday, October 02, 2025 at 09:24:21 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+4 x&=\sin \left (2 t \right )+2 t \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 26

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

\[ x = \sin \left (2 t \right ) c_2 +\cos \left (2 t \right ) c_1 -\frac {t \left (-2+\cos \left (2 t \right )\right )}{4} \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 31

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

\begin{align*} x(t)&\to \frac {t}{2}+\left (-\frac {t}{4}+c_1\right ) \cos (2 t)+c_2 \sin (2 t) \end{align*}

✓ Sympy. Time used: 0.062 (sec). Leaf size: 22

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

\[ x{\left (t \right )} = C_{2} \sin {\left (2 t \right )} + \frac {t}{2} + \left (C_{1} - \frac {t}{4}\right ) \cos {\left (2 t \right )} \]