7.12.16 problem 17

Internal problem ID [398]
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 : 17
Date solved : Tuesday, March 04, 2025 at 11:21:39 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+6 x^{\prime }+45 x&=50 \cos \left (\omega t \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 74

ode:=diff(diff(x(t),t),t)+6*diff(x(t),t)+45*x(t) = 50*cos(omega*t); 
dsolve(ode,x(t), singsol=all);
 

\[ x \left (t \right ) = \frac {\left (\omega ^{2}+12 \omega +45\right ) \left (\omega ^{2}-12 \omega +45\right ) \left (c_1 \cos \left (6 t \right )+c_2 \sin \left (6 t \right )\right ) {\mathrm e}^{-3 t}-50 \omega ^{2} \cos \left (\omega t \right )+300 \omega \sin \left (\omega t \right )+2250 \cos \left (\omega t \right )}{\omega ^{4}-54 \omega ^{2}+2025} \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 62

ode=D[x[t],{t,2}]+6*D[x[t],t]+45*x[t]==50*Cos[w*t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to -\frac {50 \left (\left (w^2-45\right ) \cos (t w)-6 w \sin (t w)\right )}{w^4-54 w^2+2025}+c_2 e^{-3 t} \cos (6 t)+c_1 e^{-3 t} \sin (6 t) \]

✓ Sympy. Time used: 0.325 (sec). Leaf size: 76

from sympy import * 
t = symbols("t") 
omega = symbols("omega") 
x = Function("x") 
ode = Eq(45*x(t) - 50*cos(omega*t) + 6*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = - \frac {50 \omega ^{2} \cos {\left (\omega t \right )}}{\omega ^{4} - 54 \omega ^{2} + 2025} + \frac {300 \omega \sin {\left (\omega t \right )}}{\omega ^{4} - 54 \omega ^{2} + 2025} + \left (C_{1} \sin {\left (6 t \right )} + C_{2} \cos {\left (6 t \right )}\right ) e^{- 3 t} + \frac {2250 \cos {\left (\omega t \right )}}{\omega ^{4} - 54 \omega ^{2} + 2025} \]