2.12.5 problem 5

Internal problem ID [912]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.6, Forced Oscillations and Resonance. Page 362
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 04:19:29 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} m x^{\prime \prime }+k x&=F_{0} \cos \left (\omega t \right ) \end{align*}

✓ Maple. Time used: 0.070 (sec). Leaf size: 60

ode:=m*diff(diff(x(t),t),t)+k*x(t) = F__0*cos(omega*t); 
dsolve(ode,x(t), singsol=all);
 

\[ x = \frac {c_1 \left (-m \,\omega ^{2}+k \right ) \cos \left (\frac {\sqrt {k}\, t}{\sqrt {m}}\right )+c_2 \left (-m \,\omega ^{2}+k \right ) \sin \left (\frac {\sqrt {k}\, t}{\sqrt {m}}\right )+F_{0} \cos \left (\omega t \right )}{-m \,\omega ^{2}+k} \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 54

ode=m*D[x[t],{t,2}]+k*x[t]==F0*Cos[omega*t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {\text {F0} \cos (\omega t)}{k-m \omega ^2}+c_1 \cos \left (\frac {\sqrt {k} t}{\sqrt {m}}\right )+c_2 \sin \left (\frac {\sqrt {k} t}{\sqrt {m}}\right ) \end{align*}

✓ Sympy. Time used: 0.069 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
F__0 = symbols("F__0") 
k = symbols("k") 
m = symbols("m") 
omega = symbols("omega") 
x = Function("x") 
ode = Eq(-F__0*cos(omega*t) + k*x(t) + m*Derivative(x(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = C_{1} e^{- t \sqrt {- \frac {k}{m}}} + C_{2} e^{t \sqrt {- \frac {k}{m}}} + \frac {F^{0} \cos {\left (\omega t \right )}}{k - m \omega ^{2}} \]