72.17.9 problem 9

Internal problem ID [14866]
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 : 9
Date solved : Thursday, March 13, 2025 at 05:15:21 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+20 y&=-3 \sin \left (2 t \right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 37

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+20*y(t) = -3*sin(2*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \sin \left (4 t \right ) {\mathrm e}^{-2 t} c_{2} +\cos \left (4 t \right ) {\mathrm e}^{-2 t} c_{1} -\frac {3 \sin \left (2 t \right )}{20}+\frac {3 \cos \left (2 t \right )}{40} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 45

ode=D[y[t],{t,2}]+4*D[y[t],t]+20*y[t]==-3*Sin[2*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {3}{40} (\cos (2 t)-2 \sin (2 t))+c_2 e^{-2 t} \cos (4 t)+c_1 e^{-2 t} \sin (4 t) \]

✓ Sympy. Time used: 0.225 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(20*y(t) + 3*sin(2*t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{1} \sin {\left (4 t \right )} + C_{2} \cos {\left (4 t \right )}\right ) e^{- 2 t} - \frac {3 \sin {\left (2 t \right )}}{20} + \frac {3 \cos {\left (2 t \right )}}{40} \]