31.1.6 problem 10.2.10

Internal problem ID [7681]
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 : Tuesday, September 30, 2025 at 04:55:52 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*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 111

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

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

✓ Mathematica. Time used: 0.312 (sec). Leaf size: 108

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

\begin{align*} 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 )} \end{align*}

✓ Sympy. Time used: 0.236 (sec). Leaf size: 129

from sympy import * 
t = symbols("t") 
F = symbols("F") 
Gamma = symbols("Gamma") 
omega = symbols("omega") 
omega__0 = symbols("omega__0") 
x = Function("x") 
ode = Eq(-F*cos(omega*t) + 2*Gamma*Derivative(x(t), t) + omega__0*x(t) + Derivative(x(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = C_{1} e^{t \left (- \Gamma + \sqrt {\Gamma ^{2} - \omega ^{0}}\right )} + C_{2} e^{- t \left (\Gamma + \sqrt {\Gamma ^{2} - \omega ^{0}}\right )} + \frac {2 F \Gamma \omega \sin {\left (\omega t \right )}}{4 \Gamma ^{2} \omega ^{2} + \omega ^{4} - 2 \omega ^{2} \omega ^{0} + \left (\omega ^{0}\right )^{2}} - \frac {F \omega ^{2} \cos {\left (\omega t \right )}}{4 \Gamma ^{2} \omega ^{2} + \omega ^{4} - 2 \omega ^{2} \omega ^{0} + \left (\omega ^{0}\right )^{2}} + \frac {F \omega ^{0} \cos {\left (\omega t \right )}}{4 \Gamma ^{2} \omega ^{2} + \omega ^{4} - 2 \omega ^{2} \omega ^{0} + \left (\omega ^{0}\right )^{2}} \]