12.1 problem 21

Internal problem ID [707]

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.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]

\[ \boxed {u^{\prime \prime }+\frac {u^{\prime }}{8}+4 u=3 \cos \left (\frac {t}{4}\right )} \] With initial conditions \begin {align*} [u \left (0\right ) = 2, u^{\prime }\left (0\right ) = 0] \end {align*}

Solution by Maple

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

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

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

Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 71

DSolve[{u''[t]+125/1000*u'[t]+4*u[t] ==3*Cos[t/4],{u[0]==0,u'[0]==0}},u[t],t,IncludeSingularSolutions -> True]
 

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