10.12.1 problem 21

Internal problem ID [1357]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 3, Second order linear equations, 3.7 Forced Vibrations. page 217
Problem number : 21
Date solved : Tuesday, March 04, 2025 at 12:30:42 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 46

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

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

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 71

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

\[ u(t)\to \frac {32 \left (1023 \sin \left (\frac {t}{4}\right )-130 \sqrt {1023} e^{-t/16} \sin \left (\frac {\sqrt {1023} t}{16}\right )+128898 \cos \left (\frac {t}{4}\right )-128898 e^{-t/16} \cos \left (\frac {\sqrt {1023} t}{16}\right )\right )}{5414057} \]

✓ Sympy. Time used: 0.295 (sec). Leaf size: 56

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

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