76.16.5 problem 20

Internal problem ID [17682]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.6 (Forced vibrations, Frequency response, and Resonance). Problems at page 272
Problem number : 20
Date solved : Tuesday, January 28, 2025 at 10:55:43 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.058 (sec). Leaf size: 46

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

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

✓ Solution by Mathematica

Time used: 0.027 (sec). Leaf size: 71

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

\[ y(t)\to \frac {98208 \sin \left (\frac {t}{4}\right )+19274 \sqrt {1023} e^{-t/16} \sin \left (\frac {\sqrt {1023} t}{16}\right )[phone] \cos \left (\frac {t}{4}\right )[phone] e^{-t/16} \cos \left (\frac {\sqrt {1023} t}{16}\right )}{16242171} \]