76.16.7 problem 22

Internal problem ID [17605]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.6 (Forced vibrations, Frequency response, and Resonance). Problems at page 272
Problem number : 22
Date solved : Thursday, March 13, 2025 at 10:42:06 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{8}+4 y&=3 \cos \left (6 t \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.048 (sec). Leaf size: 46

ode:=diff(diff(y(t),t),t)+1/8*diff(y(t),t)+4*y(t) = 3*cos(6*t); 
ic:=y(0) = 2, D(y)(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \frac {2806 \,{\mathrm e}^{-\frac {t}{16}} \sqrt {1023}\, \sin \left (\frac {\sqrt {1023}\, t}{16}\right )}{1524549}+\frac {34322 \,{\mathrm e}^{-\frac {t}{16}} \cos \left (\frac {\sqrt {1023}\, t}{16}\right )}{16393}-\frac {1536 \cos \left (6 t \right )}{16393}+\frac {36 \sin \left (6 t \right )}{16393} \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 74

ode=D[y[t],{t,2}]+125/1000*D[y[t],t]+4*y[t]==3*Cos[6*t]; 
ic={y[0]==2,Derivative[1][y][0] == 0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {e^{-t/16} \left (3348 e^{t/16} \sin (6 t)+2806 \sqrt {1023} \sin \left (\frac {\sqrt {1023} t}{16}\right )-142848 e^{t/16} \cos (6 t)+3191946 \cos \left (\frac {\sqrt {1023} t}{16}\right )\right )}{1524549} \]

✓ Sympy. Time used: 0.309 (sec). Leaf size: 56

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - 3*cos(6*t) + Derivative(y(t), t)/8 + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {2806 \sqrt {1023} \sin {\left (\frac {\sqrt {1023} t}{16} \right )}}{1524549} + \frac {34322 \cos {\left (\frac {\sqrt {1023} t}{16} \right )}}{16393}\right ) e^{- \frac {t}{16}} + \frac {36 \sin {\left (6 t \right )}}{16393} - \frac {1536 \cos {\left (6 t \right )}}{16393} \]