12.2.1 problem Problem 15.1

Internal problem ID [3484]
Book : Mathematical methods for physics and engineering, Riley, Hobson, Bence, second edition, 2002
Section : Chapter 15, Higher order ordinary differential equations. 15.4 Exercises, page 523
Problem number : Problem 15.1
Date solved : Tuesday, September 30, 2025 at 06:40:38 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+\omega _{0}^{2} x&=a \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: 0.027 (sec). Leaf size: 28

ode:=diff(diff(x(t),t),t)+omega__0^2*x(t) = a*cos(omega*t); 
ic:=[x(0) = 0, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {a \left (\cos \left (\omega _{0} t \right )-\cos \left (\omega t \right )\right )}{\omega ^{2}-\omega _{0}^{2}} \]

✓ Mathematica. Time used: 0.209 (sec). Leaf size: 33

ode=D[x[t],{t,2}]+(Subscript[\[Omega],0])^2*x[t]==a*Cos[\[Omega]*t]; 
ic={x[0]==0,Derivative[1][x][0 ]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {a \left (\cos \left (t \omega _0\right )-\cos (t \omega )\right )}{\omega ^2-\omega _0^2} \end{align*}

✓ Sympy. Time used: 0.093 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
a = symbols("a") 
omega = symbols("omega") 
omega__0 = symbols("omega__0") 
x = Function("x") 
ode = Eq(-a*cos(omega*t) + omega__0**2*x(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)
 

\[ x{\left (t \right )} = \frac {a e^{i \omega ^{0} t}}{2 \omega ^{2} - 2 \left (\omega ^{0}\right )^{2}} + \frac {a e^{- i \omega ^{0} t}}{2 \omega ^{2} - 2 \left (\omega ^{0}\right )^{2}} - \frac {a \cos {\left (\omega t \right )}}{\omega ^{2} - \left (\omega ^{0}\right )^{2}} \]