Laplace Transform of Bessel Diff Eq

[DMcG]

Hello PF, maybe you can help with this one!

I need to show that the Laplace transform of J₀(at) is (s^2 + a^2)^-1/2.

My prof told me to start with the form:
x²y'' + x y' + (x² + p²)y = 0, where p = 0 ITC.

What have I got so far?...

Doing the Laplace transform on both sides, where Y stands for L[y]:
x²(s²Y - sy(0) - y'(0)) + x(sY - y(0)) + x²Y = 0,

...then 3 lines of algebra...

Y = y(0)* (1 + xs)/(xs² + s + x) + y'(0)* x/(xs² + s + x).

Do I convolve these two terms on the RHS now? Do I factor the denominator first? Or do I use some recurrence relns for Bessel functions, which is what y(x) is?

Your help would be much appreciated!

 

[matematikawan]

Hello PF, maybe you can help with this one!

I need to show that the Laplace transform of J₀(at) is (s^2 + a^2)^-1/2.

My prof told me to start with the form:
x²y'' + x y' + (x² + p²)y = 0, where p = 0 ITC.

What have I got so far?...

Doing the Laplace transform on both sides, where Y stands for L[y]:
x²(s²Y - sy(0) - y'(0)) + x(sY - y(0)) + x²Y = 0,

...then 3 lines of algebra...

Something is not right here. I think.

L\{x^2y' ' \}=\frac{d^2}{ds^2} (s^2Y(s)- sy(0) - y'(0))

http://en.wikipedia.org/wiki/Laplace_transform"

 

[Count Iblis]

I think it is easier to start with the integral representation

J_{0}(x) =\frac{1}{2\pi}\int_{0}^{2\pi}\exp\left[i x\cos\left(\right\theta)\right]d\theta

You can then directly computethe Laplace transorm by multiplying this by exp(-sx) and then integrate over x from zero to infinity after reversing the order of integration.

It also not difficult to apply the inverse Laplace transform formula

f(x) = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\hat{f}(z)\exp(zx)dz

to the formula (s^2 + a^2)^-1/2 and then show that you get the Bessel function.