7.12.14 problem 15

Internal problem ID [396]
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 : 15
Date solved : Saturday, March 29, 2025 at 04:52:44 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 54

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

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

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 53

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

\[ x(t)\to \frac {4 w \sin (t w)-2 \left (w^2-2\right ) \cos (t w)}{w^4+4}+c_2 e^{-t} \cos (t)+c_1 e^{-t} \sin (t) \]

✓ Sympy. Time used: 0.254 (sec). Leaf size: 56

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

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