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 : Monday, January 27, 2025 at 04:54:49 AM
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*}

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 46

dsolve([diff(u(t),t$2)+125/1000*diff(u(t),t)+4*u(t) = 3*cos(6*t),u(0) = 2, D(u)(0) = 0],u(t), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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