76.16.6 problem 21

Internal problem ID [17683]
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 : 21
Date solved : Tuesday, January 28, 2025 at 10:56:58 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{8}+4 y&=3 \cos \left (2 t \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: 0.053 (sec). Leaf size: 40

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

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

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 57

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

\[ y(t)\to 12 \sin (2 t)-\frac {382 e^{-t/16} \sin \left (\frac {\sqrt {1023} t}{16}\right )}{\sqrt {1023}}+2 e^{-t/16} \cos \left (\frac {\sqrt {1023} t}{16}\right ) \]