Question on Ordinary Differential Eqn

[sero2000]

The displacement y(t) of a driven mass-spring system is described by the differential equation 3y" + 14y =7cos(2t) with initial value conditions y(0) = 0, y' (0) = 0

a.) is this system damped or un-damped?
b.) Is this system resonant?
c.) Write the solution to the IVP in terms of a product of 2 sine functions
d.) What is the frequency of the beats

Not too sure how to start on this question. Any help to jump start the thinking process would be really appreciated :D.

 

[HallsofIvy]

An obvious way to start would be to find the solution to the differential equation! Have you done that?

Do you know what "damped", "un-damped", and "resonant" mean?

 

[sero2000]

Yup i managed to get the general solution!

y(t) = 7/2 Cos(2t) + C1 Cos (*sqre root 14/3* t) + C2 Sin (*sqre root 14/3*)

Still can't find what is resonant though. does it have something to do with the sine waves?

 

[LCKurtz]

Yup i managed to get the general solution!

y(t) = 7/2 Cos(2t) + C1 Cos (*sqre root 14/3* t) + C2 Sin (*sqre root 14/3*)

Still can't find what is resonant though. does it have something to do with the sine waves?

That's correct for the general solution of the DE. But now you need to satisfy the initial conditions to get the specific solution to your problem. Then you will be ready for what's next.

 

[Rellek]

Yup i managed to get the general solution!

y(t) = 7/2 Cos(2t) + C1 Cos (*sqre root 14/3* t) + C2 Sin (*sqre root 14/3*)

Still can't find what is resonant though. does it have something to do with the sine waves?

Is resonant another word for critically damped?

Either way, if you use undetermined coefficients initially to determine the solution to your homogeneous equation, the way to tell its state of damping it to look at the discriminant for your quadratic equation.

If it's negative, it's under-damped.

If it's positive, it's over-damped.

If it's zero, it's critically damped.

 

[LCKurtz]

Still can't find what is resonant though. does it have something to do with the sine waves?

It has to do with the relationship between the forcing frequency and the natural frequency. Here's a link I think you will find very helpful, although I would expect similar information to be in your text:

http://www.marietta.edu/~mmm002/Math302/Lectures/Ch4Sec3.pdf

 

[LCKurtz]

Is resonant another word for critically damped?

No. See the link in post #6.