76.16.2 problem 16

Internal problem ID [17679]
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 : 16
Date solved : Tuesday, January 28, 2025 at 10:54:21 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{4}+2 y&=2 \cos \left (w t \right ) \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.150 (sec). Leaf size: 87

dsolve([diff(y(t),t$2)+1/4*diff(y(t),t)+2*y(t)=2*cos(w*t),y(0) = 0, D(y)(0) = 2],y(t), singsol=all)
 

\[ y = \frac {\frac {256 \sqrt {127}\, \left (w^{4}-\frac {65}{16} w^{2}+\frac {15}{4}\right ) {\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {\sqrt {127}\, t}{8}\right )}{127}+32 \,{\mathrm e}^{-\frac {t}{8}} \cos \left (\frac {\sqrt {127}\, t}{8}\right ) \left (w^{2}-2\right )-32 \cos \left (w t \right ) w^{2}+8 w \sin \left (w t \right )+64 \cos \left (w t \right )}{16 w^{4}-63 w^{2}+64} \]

✓ Solution by Mathematica

Time used: 0.052 (sec). Leaf size: 141

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

\[ y(t)\to \frac {8 e^{-t/8} \left (32 \sqrt {127} w^4 \sin \left (\frac {\sqrt {127} t}{8}\right )-130 \sqrt {127} w^2 \sin \left (\frac {\sqrt {127} t}{8}\right )+508 \left (w^2-2\right ) \cos \left (\frac {\sqrt {127} t}{8}\right )-508 e^{t/8} \left (w^2-2\right ) \cos (t w)+127 e^{t/8} w \sin (t w)+120 \sqrt {127} \sin \left (\frac {\sqrt {127} t}{8}\right )\right )}{127 \left (16 w^4-63 w^2+64\right )} \]