30.11.10 problem 10 (a)

Internal problem ID [7590]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 4, Linear Second-Order Equations. EXERCISES 4.1 at page 156
Problem number : 10 (a)
Date solved : Tuesday, September 30, 2025 at 04:54:42 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 124

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

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

✓ Mathematica. Time used: 0.05 (sec). Leaf size: 111

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

\begin{align*} y(t)&\to \frac {b \omega \sin (t \omega )+\left (k-m \omega ^2\right ) \cos (t \omega )}{b^2 \omega ^2+k^2-2 k m \omega ^2+m^2 \omega ^4}+c_1 e^{-\frac {t \left (\sqrt {b^2-4 k m}+b\right )}{2 m}}+c_2 e^{\frac {t \left (\sqrt {b^2-4 k m}-b\right )}{2 m}} \end{align*}

✓ Sympy. Time used: 0.297 (sec). Leaf size: 148

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

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