Solving integral equations, need a nudge in the right direction.

[Locoism]

Homework Statement

Solve:

\int_0^t y(τ)y(t-τ)dτ = 16sin(4t)

The Attempt at a Solution

My approach was to look at this as the convolution product y(t)*y(t), who's laplace transform should be Y(s)Y(s) = Y(s)². (Note: Maybe Fourier series are better but we haven't covered that yet).

I've tried starting by

Y(s)^2 = \frac{64}{s^2+16} and then differentiating to get

Y(s)Y'(s) = \frac{-64s}{s^2+16} which looks apealing since the inverse on the right would be 8tsin(4t). However, when I do this and take the inverse laplace, I just end up with 0=0 and I'm not making much progress...

Can anyone put me on the right track?

 

[Mute]

Laplace transform is better than Fourier transform in this case because the integral limits are from 0 to t, not the whole real line as it would need to be for the Fourier transform.

Anywho, you have an expression for $Y(s)^2$, so why not just take the square root and inverse laplace transform the square root expression, rather than trying to differentiate?

 

[Locoism]

I tried that, but what's the inverse transform of \frac{8}{\sqrt{s^2+16}}??

 

[Mute]

I tried that, but what's the inverse transform of \frac{8}{\sqrt{s^2+16}}??

Have you tried looking through a table of Laplace transforms to see if you can find one that is the inverse transform?

See the table on the wikipedia article here

Since you don't know the Fourier transform yet, I imagine you don't know how to perform an inverse laplace transform yourself, so your best bet is to try and look it up in a table. You should be able to find something you can use in the link above.

 

[Locoism]

Yea, I've looked, but all I can see is if I were to do some manipulation to make it a Bessel function, which we haven't covered either. Apart from that, I was thinking it may be multiplied by a factor e^{-16} or e^{-4} so that what we have is really Y(s-a), but it's that s² that's tripping me up. There's the possibility partial fractions, but I don't see how either. I'm sure there must be an easier way to do it...

 

[Mute]

Yea, I've looked, but all I can see is if I were to do some manipulation to make it a Bessel function, which we haven't covered either. Apart from that, I was thinking it may be multiplied by a factor e^{-16} or e^{-4} so that what we have is really Y(s-a), but it's that s² that's tripping me up. There's the possibility partial fractions, but I don't see how either. I'm sure there must be an easier way to do it...

It looks to me like your Laplace transform calculation was correct. You've found that the result may be related to a Bessel function - that is correct. I don't think you need to multiply by any exponential factors or others.

Why are you solving an integral equation if you haven't covered Fourier transforms or Bessel functions? Integral equations are an advanced topic.

 

[Locoism]

Excellent question. My teacher is a lunatic.

 

[Mute]

Excellent question. My teacher is a lunatic.

Well, that's too bad! All I can tell you is that the solution is indeed some kind of Bessel function and you don't need to muck around too much to get it. It's basically a special case of one of the results in the table on the wikipedia page.


Given that your professor seems to be teaching you things out of order, I guess you shouldn't put it past him to give you a problem with a solution which is a function you haven't studied yet?