10.12.3 problem 23

Internal problem ID [1359]
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 : 23
Date solved : Tuesday, March 04, 2025 at 12:33:16 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} u^{\prime \prime }+\frac {u^{\prime }}{8}+4 u&=3 \cos \left (6 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: 0.039 (sec). Leaf size: 46

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

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

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 74

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

\[ u(t)\to -\frac {4 e^{-t/16} \left (-3069 e^{t/16} \sin (6 t)+160 \sqrt {1023} \sin \left (\frac {\sqrt {1023} t}{16}\right )+130944 e^{t/16} \cos (6 t)-130944 \cos \left (\frac {\sqrt {1023} t}{16}\right )\right )}{5590013} \]

✓ Sympy. Time used: 0.274 (sec). Leaf size: 56

from sympy import * 
t = symbols("t") 
u = Function("u") 
ode = Eq(4*u(t) - 3*cos(6*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)
 

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