13.11.1 problem 13

Internal problem ID [2412]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.6, Mechanical Vibrations. Page 171
Problem number : 13
Date solved : Sunday, March 30, 2025 at 12:00:44 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} m y^{\prime \prime }+c y^{\prime }+k y&=F_{0} \cos \left (\omega t \right ) \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 126

ode:=m*diff(diff(y(t),t),t)+c*diff(y(t),t)+k*y(t) = F__0*cos(omega*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {F_{0} \left (-m \,\omega ^{2}+k \right ) \cos \left (\omega t \right )+F_{0} \sin \left (\omega t \right ) c \omega +\left (m^{2} \omega ^{4}+c^{2} \omega ^{2}-2 k m \,\omega ^{2}+k^{2}\right ) \left ({\mathrm e}^{\frac {\left (-c +\sqrt {c^{2}-4 k m}\right ) t}{2 m}} c_2 +{\mathrm e}^{-\frac {\left (c +\sqrt {c^{2}-4 k m}\right ) t}{2 m}} c_1 \right )}{m^{2} \omega ^{4}+c^{2} \omega ^{2}-2 k m \,\omega ^{2}+k^{2}} \]

✓ Mathematica. Time used: 0.111 (sec). Leaf size: 112

ode=m*D[y[t],{t,2}]+c*D[y[t],t]+k*y[t]==F0*Cos[\[Omega]*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {\text {F0} \left (c \omega \sin (t \omega )+\left (k-m \omega ^2\right ) \cos (t \omega )\right )}{c^2 \omega ^2+k^2-2 k m \omega ^2+m^2 \omega ^4}+c_1 e^{-\frac {t \left (\sqrt {c^2-4 k m}+c\right )}{2 m}}+c_2 e^{\frac {t \left (\sqrt {c^2-4 k m}-c\right )}{2 m}} \]

✓ Sympy. Time used: 0.510 (sec). Leaf size: 153

from sympy import * 
t = symbols("t") 
F__0 = symbols("F__0") 
c = symbols("c") 
k = symbols("k") 
m = symbols("m") 
omega = symbols("omega") 
y = Function("y") 
ode = Eq(-F__0*cos(omega*t) + c*Derivative(y(t), t) + k*y(t) + m*Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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