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I need to show that the Laplace transform of J

_{0}(at) is (s^2 + a^2)^-1/2.

My prof told me to start with the form:

x

^{2}y'' + x y' + (x

^{2}+ p

^{2})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]:

x

^{2}(s

^{2}Y - sy(0) - y'(0)) + x(sY - y(0)) + x

^{2}Y = 0,

...then 3 lines of algebra...

Y = y(0)* (1 + xs)/(xs

^{2}+ s + x) + y'(0)* x/(xs

^{2}+ 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!!