66.17.12 problem 12

Internal problem ID [16235]
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 : 12
Date solved : Thursday, October 02, 2025 at 10:43:56 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+8 y&=2 \cos \left (3 t \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.035 (sec). Leaf size: 29

ode:=diff(diff(y(t),t),t)+6*diff(y(t),t)+8*y(t) = 2*cos(3*t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {4 \,{\mathrm e}^{-4 t}}{25}-\frac {2 \,{\mathrm e}^{-2 t}}{13}+\frac {36 \sin \left (3 t \right )}{325}-\frac {2 \cos \left (3 t \right )}{325} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 74

ode=D[y[t],{t,2}]+5*D[y[t],t]+8*y[t]==2*Cos[3*t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{791} e^{-5 t/2} \left (105 e^{5 t/2} \sin (3 t)-85 \sqrt {7} \sin \left (\frac {\sqrt {7} t}{2}\right )-7 e^{5 t/2} \cos (3 t)+7 \cos \left (\frac {\sqrt {7} t}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.159 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(8*y(t) - 2*cos(3*t) + 6*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {36 \sin {\left (3 t \right )}}{325} - \frac {2 \cos {\left (3 t \right )}}{325} - \frac {2 e^{- 2 t}}{13} + \frac {4 e^{- 4 t}}{25} \]