Internal problem ID [4542]
Book: Basic Training in Mathematics. By R. Shankar. Plenum Press. NY. 1995
Section: Chapter 10, Differential equations. Section 10.2, ODEs with constant Coefficients. page
307
Problem number: 10.2.10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {x^{\prime \prime }+2 \gamma x^{\prime }+\omega _{0} x-F \cos \left (\omega t \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 89
dsolve(diff(x(t),t$2)+2*gamma*diff(x(t),t)+omega__0*x(t)=F*cos(omega*t),x(t), singsol=all)
\[ x \relax (t ) = {\mathrm e}^{\left (-\gamma +\sqrt {\gamma ^{2}-\omega _{0}}\right ) t} c_{2}+{\mathrm e}^{\left (-\gamma -\sqrt {\gamma ^{2}-\omega _{0}}\right ) t} c_{1}+\frac {F \left (-\omega ^{2}+\omega _{0}\right ) \cos \left (\omega t \right )+2 F \sin \left (\omega t \right ) \gamma \omega }{\omega ^{4}+\left (4 \gamma ^{2}-2 \omega _{0}\right ) \omega ^{2}+\omega _{0}^{2}} \]
✓ Solution by Mathematica
Time used: 0.222 (sec). Leaf size: 106
DSolve[x''[t]+2*\[Gamma]*x'[t]+Subscript[\[Omega],0]*x[t]==F*Cos[\[Omega]*t],x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {2 \gamma F \omega \sin (t \omega )+F \left (\omega _0-\omega ^2\right ) \cos (t \omega )}{4 \gamma ^2 \omega ^2+\omega ^4-2 \omega _0 \omega ^2+\omega _0^2}+c_1 e^{-t \left (\sqrt {\gamma ^2-\omega _0}+\gamma \right )}+c_2 e^{t \left (\sqrt {\gamma ^2-\omega _0}-\gamma \right )} \\ \end{align*}