10.12.2 problem 22

Internal problem ID [1358]
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 : 22
Date solved : Saturday, March 29, 2025 at 10:53:15 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} u^{\prime \prime }+\frac {u^{\prime }}{8}+4 u&=3 \cos \left (2 t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 40

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

\[ u = -\frac {382 \,{\mathrm e}^{-\frac {t}{16}} \sqrt {1023}\, \sin \left (\frac {\sqrt {1023}\, t}{16}\right )}{1023}+2 \,{\mathrm e}^{-\frac {t}{16}} \cos \left (\frac {\sqrt {1023}\, t}{16}\right )+12 \sin \left (2 t \right ) \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 39

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

\[ u(t)\to 12 \sin (2 t)-128 \sqrt {\frac {3}{341}} e^{-t/16} \sin \left (\frac {\sqrt {1023} t}{16}\right ) \]

✓ Sympy. Time used: 0.270 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
u = Function("u") 
ode = Eq(4*u(t) - 3*cos(2*t) + 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)
 

\[ u{\left (t \right )} = \left (- \frac {382 \sqrt {1023} \sin {\left (\frac {\sqrt {1023} t}{16} \right )}}{1023} + 2 \cos {\left (\frac {\sqrt {1023} t}{16} \right )}\right ) e^{- \frac {t}{16}} + 12 \sin {\left (2 t \right )} \]