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
:
Sunday, March 30, 2025 at 10:54:17 AM
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: 0.502 (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]
\[
x(t)\to \frac {F \left (\omega (2 \gamma \sin (t \omega )-\omega \cos (t \omega ))+\omega _0 \cos (t \omega )\right )}{4 \gamma ^2 \omega ^2+\omega ^4-2 \omega _0 \omega ^2+\omega _0^2}+c_1 e^{-t \left (\sqrt {\gamma ^2-\omega _0}+\gamma \right )}+c_2 e^{t \left (\sqrt {\gamma ^2-\omega _0}-\gamma \right )}
\]
✓ Sympy. Time used: 0.373 (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)
\[
x{\left (t \right )} = C_{1} e^{t \left (- \Gamma + \sqrt {\Gamma ^{2} - \omega ^{0}}\right )} + C_{2} e^{- t \left (\Gamma + \sqrt {\Gamma ^{2} - \omega ^{0}}\right )} + \frac {2 F \Gamma \omega \sin {\left (\omega t \right )}}{4 \Gamma ^{2} \omega ^{2} + \omega ^{4} - 2 \omega ^{2} \omega ^{0} + \left (\omega ^{0}\right )^{2}} - \frac {F \omega ^{2} \cos {\left (\omega t \right )}}{4 \Gamma ^{2} \omega ^{2} + \omega ^{4} - 2 \omega ^{2} \omega ^{0} + \left (\omega ^{0}\right )^{2}} + \frac {F \omega ^{0} \cos {\left (\omega t \right )}}{4 \Gamma ^{2} \omega ^{2} + \omega ^{4} - 2 \omega ^{2} \omega ^{0} + \left (\omega ^{0}\right )^{2}}
\]