66.17.8 problem 8

Internal problem ID [16231]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 4. Forcing and Resonance. Section 4.2 page 412
Problem number : 8
Date solved : Thursday, October 02, 2025 at 10:43:53 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+20 y&=-\cos \left (5 t \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 37

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+20*y(t) = -cos(5*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{-2 t} \sin \left (4 t \right ) c_2 +{\mathrm e}^{-2 t} \cos \left (4 t \right ) c_1 -\frac {4 \sin \left (5 t \right )}{85}+\frac {\cos \left (5 t \right )}{85} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 45

ode=D[y[t],{t,2}]+4*D[y[t],t]+20*y[t]==-Cos[5*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{85} (\cos (5 t)-4 \sin (5 t))+c_2 e^{-2 t} \cos (4 t)+c_1 e^{-2 t} \sin (4 t) \end{align*}

✓ Sympy. Time used: 0.150 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(20*y(t) + cos(5*t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{1} \sin {\left (4 t \right )} + C_{2} \cos {\left (4 t \right )}\right ) e^{- 2 t} - \frac {4 \sin {\left (5 t \right )}}{85} + \frac {\cos {\left (5 t \right )}}{85} \]