7.12.17 problem 18

Internal problem ID [399]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 2. Linear Equations of Higher Order. Section 2.6 (Forced oscillations and resonance). Problems at page 171
Problem number : 18
Date solved : Saturday, March 29, 2025 at 04:52:48 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+10 x^{\prime }+650 x&=100 \cos \left (\omega t \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 58

ode:=diff(diff(x(t),t),t)+10*diff(x(t),t)+650*x(t) = 100*cos(omega*t); 
dsolve(ode,x(t), singsol=all);
 

\[ x = {\mathrm e}^{-5 t} \sin \left (25 t \right ) c_2 +{\mathrm e}^{-5 t} \cos \left (25 t \right ) c_1 +\frac {\left (-100 \omega ^{2}+65000\right ) \cos \left (\omega t \right )+1000 \omega \sin \left (\omega t \right )}{\omega ^{4}-1200 \omega ^{2}+422500} \]

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 62

ode=D[x[t],{t,2}]+10*D[x[t],t]+650*x[t]==100*Cos[w*t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to -\frac {100 \left (\left (w^2-650\right ) \cos (t w)-10 w \sin (t w)\right )}{w^4-1200 w^2+422500}+c_2 e^{-5 t} \cos (25 t)+c_1 e^{-5 t} \sin (25 t) \]

✓ Sympy. Time used: 0.326 (sec). Leaf size: 76

from sympy import * 
t = symbols("t") 
omega = symbols("omega") 
x = Function("x") 
ode = Eq(650*x(t) - 100*cos(omega*t) + 10*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = - \frac {100 \omega ^{2} \cos {\left (\omega t \right )}}{\omega ^{4} - 1200 \omega ^{2} + 422500} + \frac {1000 \omega \sin {\left (\omega t \right )}}{\omega ^{4} - 1200 \omega ^{2} + 422500} + \left (C_{1} \sin {\left (25 t \right )} + C_{2} \cos {\left (25 t \right )}\right ) e^{- 5 t} + \frac {65000 \cos {\left (\omega t \right )}}{\omega ^{4} - 1200 \omega ^{2} + 422500} \]