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]]
Solve \begin {gather*} \boxed {m y^{\prime \prime }+c y^{\prime }+y k -F_{0} \cos \left (\omega t \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.013 (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)
\[ y \relax (t ) = {\mathrm e}^{\frac {\left (-c +\sqrt {c^{2}-4 k m}\right ) t}{2 m}} c_{2}+{\mathrm e}^{-\frac {\left (c +\sqrt {c^{2}-4 k m}\right ) t}{2 m}} c_{1}+\frac {F_{0} \left (\sin \left (\omega t \right ) c \omega +\left (-m \,\omega ^{2}+k \right ) \cos \left (\omega t \right )\right )}{m^{2} \omega ^{4}+\left (c^{2}-2 k m \right ) \omega ^{2}+k^{2}} \]
✓ Solution by Mathematica
Time used: 0.332 (sec). Leaf size: 100
DSolve[m*y''[t]+c*y'[t]+k*y[t]==F0*Cos[\[Omega]*t],y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {c \text {F0} \omega \sin (t \omega )+\text {F0} \left (k-m \omega ^2\right ) \cos (t \omega )}{c^2 \omega ^2+\left (k-m \omega ^2\right )^2}+e^{-\frac {t \left (\sqrt {c^2-4 k m}+c\right )}{2 m}} \left (c_2 e^{\frac {t \sqrt {c^2-4 k m}}{m}}+c_1\right ) \\ \end{align*}