Laplace Transform of a Bessel Equation

[wtmoore]

Hi guys, I have this question on Laplace transforms, but am not sure how to start it.

The zero order Bessel function Jo(t) satisfies the ordinary differential equation:
tJ''o(t) + J'o(t) + tJo(t) = 0

Take the Laplace transform of this equation and use the properties
of the transform to find the Laplace transform of J0(at) where a is a constant.

Now I can do laplace transforms, but I haven't been given it in this form before, and am not too sure on Bessel functions. I have googled, and am having trouble finding out exactly what J(t) is. I am going to have 3 separate integrals right? Do you know what Jo(t) is?

Thanks

 

[lippyka]

!Yes, you will have three separate integrals. Jo(t) is the zero-order Bessel function of the first kind. It is a special function that is used in many different fields and is sometimes referred to as an oscillatory or cylindrical function. The equation you have been given is the differential equation that defines Jo(t). To take the Laplace transform of the equation, you need to first rewrite it as an integral equation by integrating both sides of the original equation with respect to t. This should give you:∫ 0 ∞ [tJ''o(t) + J'o(t) + tJo(t)]dt = 0You can then apply the Laplace transform to this integral equation, taking care to use the properties of the Laplace transform. Once you have found the Laplace transform of the equation, you can use this to find the Laplace transform of J0(at).