65.7.7 problem 14.1 (vii)

Internal problem ID [13692]
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 (vii)
Date solved : Wednesday, March 05, 2025 at 10:12:15 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 36

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

\[ x \left (t \right ) = \frac {\sin \left (\omega t \right ) \left (2 c_{2} \omega ^{2}+1\right )-\omega \cos \left (\omega t \right ) \left (-2 c_{1} \omega +t \right )}{2 \omega ^{2}} \]

✓ Mathematica. Time used: 0.052 (sec). Leaf size: 61

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

\[ x(t)\to \cos (t w) \int _1^t-\frac {\sin ^2(w K[1])}{w}dK[1]-\frac {\sin (t w) \cos ^2(t w)}{2 w^2}+c_1 \cos (t w)+c_2 \sin (t w) \]

✓ Sympy. Time used: 0.129 (sec). Leaf size: 29

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

\[ x{\left (t \right )} = C_{1} e^{- i \omega t} + C_{2} e^{i \omega t} - \frac {t \cos {\left (\omega t \right )}}{2 \omega } \]