30.11.11 problem 10 (b)

Internal problem ID [7591]
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 (b)
Date solved : Tuesday, September 30, 2025 at 04:54:43 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{10}+25 y&=\cos \left (\omega t \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 90

ode:=diff(diff(y(t),t),t)+1/10*diff(y(t),t)+25*y(t) = cos(omega*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {100 c_1 \,{\mathrm e}^{-\frac {t}{20}} \left (\omega ^{4}-\frac {4999}{100} \omega ^{2}+625\right ) \cos \left (\frac {3 \sqrt {1111}\, t}{20}\right )+100 c_2 \,{\mathrm e}^{-\frac {t}{20}} \left (\omega ^{4}-\frac {4999}{100} \omega ^{2}+625\right ) \sin \left (\frac {3 \sqrt {1111}\, t}{20}\right )-100 \cos \left (\omega t \right ) \omega ^{2}+10 \sin \left (\omega t \right ) \omega +2500 \cos \left (\omega t \right )}{100 \omega ^{4}-4999 \omega ^{2}+62500} \]

✓ Mathematica. Time used: 1.03 (sec). Leaf size: 137

ode=1*D[y[t],{t,2}]+1/10*D[y[t],t]+25*y[t]==Cos[\[Omega]*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {e^{-t/20} \left (100 c_1 \omega ^4 \sin \left (\frac {3 \sqrt {1111} t}{20}\right )-100 e^{t/20} \left (\omega ^2-25\right ) \cos (t \omega )-4999 c_1 \omega ^2 \sin \left (\frac {3 \sqrt {1111} t}{20}\right )+c_2 \left (100 \omega ^4-4999 \omega ^2+62500\right ) \cos \left (\frac {3 \sqrt {1111} t}{20}\right )+10 e^{t/20} \omega \sin (t \omega )+62500 c_1 \sin \left (\frac {3 \sqrt {1111} t}{20}\right )\right )}{100 \omega ^4-4999 \omega ^2+62500} \end{align*}

✓ Sympy. Time used: 0.207 (sec). Leaf size: 95

from sympy import * 
t = symbols("t") 
w = symbols("w") 
y = Function("y") 
ode = Eq(25*y(t) - cos(t*w) + Derivative(y(t), t)/10 + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - \frac {100 w^{2} \cos {\left (t w \right )}}{100 w^{4} - 4999 w^{2} + 62500} + \frac {10 w \sin {\left (t w \right )}}{100 w^{4} - 4999 w^{2} + 62500} + \left (C_{1} \sin {\left (\frac {3 \sqrt {1111} t}{20} \right )} + C_{2} \cos {\left (\frac {3 \sqrt {1111} t}{20} \right )}\right ) e^{- \frac {t}{20}} + \frac {2500 \cos {\left (t w \right )}}{100 w^{4} - 4999 w^{2} + 62500} \]