13.11.1 problem 13

Internal problem ID [2412]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.6, Mechanical Vibrations. Page 171
Problem number : 13
Date solved : Monday, January 27, 2025 at 05:52:02 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 126

dsolve(m*diff(y(t),t$2)+c*diff(y(t),t)+k*y(t)=F__0*cos(omega*t),y(t), singsol=all)
 

\[ y = \frac {F_{0} \left (-m \,\omega ^{2}+k \right ) \cos \left (\omega t \right )+F_{0} \sin \left (\omega t \right ) c \omega +\left (m^{2} \omega ^{4}+c^{2} \omega ^{2}-2 k m \,\omega ^{2}+k^{2}\right ) \left ({\mathrm e}^{\frac {\left (-c +\sqrt {c^{2}-4 k m}\right ) t}{2 m}} c_2 +{\mathrm e}^{-\frac {\left (c +\sqrt {c^{2}-4 k m}\right ) t}{2 m}} c_1 \right )}{m^{2} \omega ^{4}+c^{2} \omega ^{2}-2 k m \,\omega ^{2}+k^{2}} \]

✓ Solution by Mathematica

Time used: 0.111 (sec). Leaf size: 112

DSolve[m*D[y[t],{t,2}]+c*D[y[t],t]+k*y[t]==F0*Cos[\[Omega]*t],y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {\text {F0} \left (c \omega \sin (t \omega )+\left (k-m \omega ^2\right ) \cos (t \omega )\right )}{c^2 \omega ^2+k^2-2 k m \omega ^2+m^2 \omega ^4}+c_1 e^{-\frac {t \left (\sqrt {c^2-4 k m}+c\right )}{2 m}}+c_2 e^{\frac {t \left (\sqrt {c^2-4 k m}-c\right )}{2 m}} \]