72.17.11 problem 11

Internal problem ID [14868]
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 : 11
Date solved : Thursday, March 13, 2025 at 05:15:52 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+8 y&=\cos \left (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: 0.024 (sec). Leaf size: 25

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

\[ y = \frac {2 \,{\mathrm e}^{-4 t}}{17}+\frac {7 \cos \left (t \right )}{85}+\frac {6 \sin \left (t \right )}{85}-\frac {{\mathrm e}^{-2 t}}{5} \]

✓ Mathematica. Time used: 2.029 (sec). Leaf size: 202

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

\[ y(t)\to e^{-5 t/2} \left (-\sin \left (\frac {\sqrt {7} t}{2}\right ) \int _1^0\frac {2 e^{\frac {5 K[1]}{2}} \cos (K[1]) \cos \left (\frac {1}{2} \sqrt {7} K[1]\right )}{\sqrt {7}}dK[1]+\sin \left (\frac {\sqrt {7} t}{2}\right ) \int _1^t\frac {2 e^{\frac {5 K[1]}{2}} \cos (K[1]) \cos \left (\frac {1}{2} \sqrt {7} K[1]\right )}{\sqrt {7}}dK[1]+\cos \left (\frac {\sqrt {7} t}{2}\right ) \left (\int _1^t-\frac {2 e^{\frac {5 K[2]}{2}} \cos (K[2]) \sin \left (\frac {1}{2} \sqrt {7} K[2]\right )}{\sqrt {7}}dK[2]-\int _1^0-\frac {2 e^{\frac {5 K[2]}{2}} \cos (K[2]) \sin \left (\frac {1}{2} \sqrt {7} K[2]\right )}{\sqrt {7}}dK[2]\right )\right ) \]

✓ Sympy. Time used: 0.226 (sec). Leaf size: 32

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 = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {6 \sin {\left (t \right )}}{85} + \frac {7 \cos {\left (t \right )}}{85} - \frac {e^{- 2 t}}{5} + \frac {2 e^{- 4 t}}{17} \]