72.17.17 problem 19

Internal problem ID [14874]
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 : 19
Date solved : Thursday, March 13, 2025 at 05:17:22 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 36

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

\[ y = \left (\cos \left (4 t \right ) c_{1} +\sin \left (4 t \right ) c_{2} \right ) {\mathrm e}^{-2 t}+\frac {4 \left (\cos \left (t \right )+\frac {\sin \left (t \right )}{8}\right ) {\mathrm e}^{-t}}{65} \]

✓ Mathematica. Time used: 0.206 (sec). Leaf size: 82

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

\[ y(t)\to e^{-2 t} \left (\cos (4 t) \int _1^t-\frac {1}{4} e^{K[2]} \cos (K[2]) \sin (4 K[2])dK[2]+\sin (4 t) \int _1^t\frac {1}{4} e^{K[1]} \cos (K[1]) \cos (4 K[1])dK[1]+c_2 \cos (4 t)+c_1 \sin (4 t)\right ) \]

✓ Sympy. Time used: 0.324 (sec). Leaf size: 34

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

\[ y{\left (t \right )} = \left (\left (C_{1} \sin {\left (4 t \right )} + C_{2} \cos {\left (4 t \right )}\right ) e^{- t} + \frac {\sin {\left (t \right )}}{130} + \frac {4 \cos {\left (t \right )}}{65}\right ) e^{- t} \]