76.16.4 problem 18

Internal problem ID [17681]
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 : 18
Date solved : Tuesday, January 28, 2025 at 10:55:40 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 36

dsolve([diff(y(t),t$2)+y(t)=3*cos(w*t),y(0) = 2, D(y)(0) = 1],y(t), singsol=all)
 

\[ y = \frac {-3 \cos \left (w t \right )+\left (2 \cos \left (t \right )+\sin \left (t \right )\right ) w^{2}+\cos \left (t \right )-\sin \left (t \right )}{w^{2}-1} \]

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 36

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

\[ y(t)\to \frac {\left (w^2-1\right ) \sin (t)+\left (w^2+2\right ) \cos (t)-3 \cos (t w)}{w^2-1} \]