66.17.5 problem 5

Internal problem ID [16228]
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 : 5
Date solved : Thursday, October 02, 2025 at 10:43:51 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 26

ode:=diff(diff(y(t),t),t)+6*diff(y(t),t)+8*y(t) = cos(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{-4 t} c_1}{2}+\frac {7 \cos \left (t \right )}{85}+\frac {6 \sin \left (t \right )}{85}+{\mathrm e}^{-2 t} c_2 \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 68

ode=D[y[t],{t,2}]+6*D[y[t],t]+8*y[t]==Cos[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.125 (sec). Leaf size: 29

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

\[ y{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} e^{- 2 t} + \frac {6 \sin {\left (t \right )}}{85} + \frac {7 \cos {\left (t \right )}}{85} \]