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 : Monday, January 27, 2025 at 04:53:32 AM
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*}

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 40

dsolve([diff(u(t),t$2)+125/1000*diff(u(t),t)+4*u(t) = 3*cos(2*t),u(0) = 2, D(u)(0) = 0],u(t), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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