65.8.2 problem 15.3

Internal problem ID [13700]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 15, Resonance. Exercises page 148
Problem number : 15.3
Date solved : Wednesday, March 05, 2025 at 10:13:14 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 14

ode:=diff(diff(x(t),t),t)+omega^2*x(t) = cos(omega*t); 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x \left (t \right ) = \frac {\sin \left (\omega t \right ) t}{2 \omega } \]

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 68

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

\[ x(t)\to -\frac {\sin (t w) \left (-4 w^2 \int _1^t\frac {\cos ^2(w K[1])}{w}dK[1]+4 w^2 \int _1^0\frac {\cos ^2(w K[1])}{w}dK[1]+\sin (2 t w)\right )}{4 w^2} \]

✓ Sympy. Time used: 0.145 (sec). Leaf size: 12

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

\[ x{\left (t \right )} = \frac {t \sin {\left (\omega t \right )}}{2 \omega } \]