66.17.15 problem 15

Internal problem ID [16238]
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 : 15
Date solved : Thursday, October 02, 2025 at 10:43:58 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+y&=\cos \left (3 t \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 39

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t)+y(t) = cos(3*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{\frac {\left (\sqrt {5}-3\right ) t}{2}} c_2 +{\mathrm e}^{-\frac {\left (3+\sqrt {5}\right ) t}{2}} c_1 +\frac {9 \sin \left (3 t \right )}{145}-\frac {8 \cos \left (3 t \right )}{145} \]

✓ Mathematica. Time used: 0.181 (sec). Leaf size: 112

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

\begin{align*} y(t)&\to e^{-\frac {1}{2} \left (3+\sqrt {5}\right ) t} \left (\int _1^t-\frac {e^{\frac {1}{2} \left (3+\sqrt {5}\right ) K[1]} \cos (3 K[1])}{\sqrt {5}}dK[1]+e^{\sqrt {5} t} \int _1^t\frac {e^{-\frac {1}{2} \left (-3+\sqrt {5}\right ) K[2]} \cos (3 K[2])}{\sqrt {5}}dK[2]+c_2 e^{\sqrt {5} t}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.169 (sec). Leaf size: 46

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

\[ y{\left (t \right )} = C_{1} e^{\frac {t \left (-3 + \sqrt {5}\right )}{2}} + C_{2} e^{- \frac {t \left (\sqrt {5} + 3\right )}{2}} + \frac {9 \sin {\left (3 t \right )}}{145} - \frac {8 \cos {\left (3 t \right )}}{145} \]