37.1.6 problem 10.2.10

Internal problem ID [6393]
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
Date solved : Monday, January 27, 2025 at 01:58:47 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+2 \gamma x^{\prime }+\omega _{0} x&=F \cos \left (\omega t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 111

dsolve(diff(x(t),t$2)+2*gamma*diff(x(t),t)+omega__0*x(t)=F*cos(omega*t),x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {-F \left (\omega ^{2}-\omega _{0} \right ) \cos \left (\omega t \right )+2 F \sin \left (\omega t \right ) \gamma \omega +4 \left ({\mathrm e}^{-\left (\gamma +\sqrt {\gamma ^{2}-\omega _{0}}\right ) t} c_1 +{\mathrm e}^{\left (-\gamma +\sqrt {\gamma ^{2}-\omega _{0}}\right ) t} c_2 \right ) \left (\frac {\omega ^{4}}{4}+\left (\gamma ^{2}-\frac {\omega _{0}}{2}\right ) \omega ^{2}+\frac {\omega _{0}^{2}}{4}\right )}{\omega ^{4}+\left (4 \gamma ^{2}-2 \omega _{0} \right ) \omega ^{2}+\omega _{0}^{2}} \]

✓ Solution by Mathematica

Time used: 0.502 (sec). Leaf size: 108

DSolve[D[x[t],{t,2}]+2*\[Gamma]*D[x[t],t]+Subscript[\[Omega],0]*x[t]==F*Cos[\[Omega]*t],x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \frac {F \left (\omega (2 \gamma \sin (t \omega )-\omega \cos (t \omega ))+\omega _0 \cos (t \omega )\right )}{4 \gamma ^2 \omega ^2+\omega ^4-2 \omega _0 \omega ^2+\omega _0^2}+c_1 e^{-t \left (\sqrt {\gamma ^2-\omega _0}+\gamma \right )}+c_2 e^{t \left (\sqrt {\gamma ^2-\omega _0}-\gamma \right )} \]