76.16.5 problem 20

Internal problem ID [17611]
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 : 20
Date solved : Monday, March 31, 2025 at 04:22:30 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 46

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

\[ y = \frac {19274 \,{\mathrm e}^{-\frac {t}{16}} \sqrt {1023}\, \sin \left (\frac {\sqrt {1023}\, t}{16}\right )}{16242171}+\frac {19658 \,{\mathrm e}^{-\frac {t}{16}} \cos \left (\frac {\sqrt {1023}\, t}{16}\right )}{15877}+\frac {96 \sin \left (\frac {t}{4}\right )}{15877}+\frac {12096 \cos \left (\frac {t}{4}\right )}{15877} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 71

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

\[ y(t)\to \frac {98208 \sin \left (\frac {t}{4}\right )+19274 \sqrt {1023} e^{-t/16} \sin \left (\frac {\sqrt {1023} t}{16}\right )[phone] \cos \left (\frac {t}{4}\right )[phone] e^{-t/16} \cos \left (\frac {\sqrt {1023} t}{16}\right )}{16242171} \]

✓ Sympy. Time used: 0.275 (sec). Leaf size: 56

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - 3*cos(t/4) + 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)
 

\[ y{\left (t \right )} = \left (\frac {19274 \sqrt {1023} \sin {\left (\frac {\sqrt {1023} t}{16} \right )}}{16242171} + \frac {19658 \cos {\left (\frac {\sqrt {1023} t}{16} \right )}}{15877}\right ) e^{- \frac {t}{16}} + \frac {96 \sin {\left (\frac {t}{4} \right )}}{15877} + \frac {12096 \cos {\left (\frac {t}{4} \right )}}{15877} \]