Laplace transform and bessels equation

[wtmoore]

Homework Statement

I'm trying to find the Laplace transform of tJ''₀(t), it's from bessels equation, but that doesn't matter too much at the moment, I just need to integrate (e^-st)*t*J''₀(t) but am unsure how to go about this with the J''₀(t) in there.

 

[Ray Vickson]

Maple 14 gets an answer:
t*diff(BesselJ(0,t),t$2): <---second derivative of J0(t), times t

f:=expand(%);
f := -t BesselJ(0, t) + BesselJ(1, t)

L:=int(f*exp(-s*t),t=0..infinity);

L:= [1 + 2s^2 + s^4 - (s^2 + 2s)*sqrt(1 + s^2)]/ (1+s^2)^2

Note: Maple used the DE for J0 and recursions to express J0'' in terms of J0 and J1, then it integrated that.

RGV