7.12.5 problem 5

Internal problem ID [387]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 2. Linear Equations of Higher Order. Section 2.6 (Forced oscillations and resonance). Problems at page 171
Problem number : 5
Date solved : Saturday, March 29, 2025 at 04:52:27 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} m x^{\prime \prime }+k x&=F_{0} \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: 0.036 (sec). Leaf size: 32

ode:=m*diff(diff(x(t),t),t)+k*x(t) = F__0*cos(omega*t); 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 48

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

\[ x(t)\to \frac {f \left (\sin (t w)-\frac {\sqrt {m} w \sin \left (\frac {\sqrt {k} t}{\sqrt {m}}\right )}{\sqrt {k}}\right )}{k-m w^2} \]

✓ Sympy. Time used: 0.169 (sec). Leaf size: 63

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 = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

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