Math 238	Exploring Forced Vibrations
The following manipulate can be used to explore the equation 2u''+u'+4u=2cos(ω t).  The last two lines of manipulate command create two interactive sliders:  

	v1 has an initial value of 1, and a range of values between 0 and 2, and increments by 0.01.
	xmax has an initial value of 10 with values between 5 and 100.
	
v1 changes the frequency ω in the forcing function, and xmax changes the domain for the numeric solution and the plot window. (You can add as many sliders as needed.)

Module is a “program” in Mathematica.  For this one, sol is a local variable, and the two commands are separated with a semicolon.

Manipulate[
Module[{sol},
sol=u/.First@NDSolve[{2 u''[t]+ u'[t]+4u[t]==2Cos[v1 t],u[0]==0,u'[0]==0},u,{t,0,xmax}];
Plot[sol[t],{t,0,xmax},PlotRange->{-2,2}]
],
{{v1,1},0,2,.01,Appearance->"Labeled"},
{{xmax,10},5,100}
]

1)	Estimate the amplitude (Subscript[A, s]) of the steady state solution when ω=1.

2)	Find the value of ω that maximizes (Subscript[A, s]).

3)	Create a slider to change the damping coefficient γ in the equation and repeat problem 2 using the values γ=0.25, γ=0.5, and γ=0.75.

4)	Make a plot of the points (ω,(Subscript[A, s])) and graph (Subscript[A, s])vs ω.  How does the maximum value of (Subscript[A, s]) vary with ω?  Compare your results with equation (7) on page 208.