Internal problem ID [1764]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.6, Mechanical Vibrations. Page 171
Problem number: 13.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {m y^{\prime \prime }+c y^{\prime }+k y=F_{0} \cos \left (\omega t \right )} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 97
dsolve(m*diff(y(t),t$2)+c*diff(y(t),t)+k*y(t)=F__0*cos(omega*t),y(t), singsol=all)
\[ y \left (t \right ) = {\mathrm e}^{\frac {\left (-c +\sqrt {c^{2}-4 k m}\right ) t}{2 m}} c_{2} +{\mathrm e}^{-\frac {\left (c +\sqrt {c^{2}-4 k m}\right ) t}{2 m}} c_{1} +\frac {F_{0} \left (\left (-m \,\omega ^{2}+k \right ) \cos \left (\omega t \right )+\sin \left (\omega t \right ) c \omega \right )}{m^{2} \omega ^{4}+\left (c^{2}-2 k m \right ) \omega ^{2}+k^{2}} \]
✓ Solution by Mathematica
Time used: 0.119 (sec). Leaf size: 112
DSolve[m*y''[t]+c*y'[t]+k*y[t]==F0*Cos[\[Omega]*t],y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {\text {F0} \left (c \omega \sin (t \omega )+\left (k-m \omega ^2\right ) \cos (t \omega )\right )}{c^2 \omega ^2+k^2-2 k m \omega ^2+m^2 \omega ^4}+c_1 e^{-\frac {t \left (\sqrt {c^2-4 k m}+c\right )}{2 m}}+c_2 e^{\frac {t \left (\sqrt {c^2-4 k m}-c\right )}{2 m}} \]