76.16.4 problem 18

Internal problem ID [17610]
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 : Monday, March 31, 2025 at 04:22:29 PM
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*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 36

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

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

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

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

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

✓ Sympy. Time used: 0.097 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
w = symbols("w") 
y = Function("y") 
ode = Eq(y(t) - 3*cos(t*w) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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