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 : Monday, January 27, 2025 at 02:48:21 AM
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*}

✓ Solution by Maple

Time used: 0.028 (sec). Leaf size: 32

dsolve([m*diff(x(t),t$2)+k*x(t)=F__0*cos(omega*t),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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