# Help With a Laplace Transform

## Homework Statement

$$xy''+y'+xy=0, y'(0)=0, y(0)=0$$ Using the method of Laplace transforms, show that the solution is the Bessel function of order zero.

## Homework Equations

$$-(d/ds)L{f(x)}$$

## The Attempt at a Solution

The only thing I got out of this when trying to solve it was y=0. Obviously not the intended answer. In the problem I'm told that the answer is the Bessel function of 0 order.

I'm pretty sure the parts I'm messing up is the Laplace transform of xy''. I haven't tried to take the Laplace transform of anything like that before and I'm sure it is where I'm messing up. I tried to use the relevant equation up there and I got...

$$-(d/ds)[(s^2Y(s)-s)]+sY(s)-1+-(d/ds)[Y(s)]=0$$

This gives me $$-2sY(s)+1+sY(s)-1=0$$

=> $$-sY(s)=0$$

which yeah, just gives me a dumb answer. Again I'm pretty sure it's with my xy'' term. Maybe even more :(

Thanks anyone for help.

Oh... as soon as I posted this I thought... Y(s) is a function of s... I need to take that into account when doing derivatives... Dang it... as always it's right after I post I have a decent idea...

Last edited:
Ok so I'm on the right track for sure. My current expression is

$$y = L^-1[1/(s^2+1)^.5]$$

I know that the inverse of that is the zero order Bessel function. How would I prove with no table though?

SteamKing
Staff Emeritus
Homework Helper
Ok so I'm on the right track for sure. My current expression is

$$y = L^-1[1/(s^2+1)^.5]$$

I know that the inverse of that is the zero order Bessel function. How would I prove with no table though?
Well fortunately, there are tables of the Laplace transforms of all kinds of special functions, including Bessel functions. One reference which has such a table is Abramowitz & Stegun, Handbook of Mathematical Functions, p. 1029:

http://people.math.sfu.ca/~cbm/aands/

If you don't trust such a reference, I suppose you could always derive the Laplace transform of your Bessel function from the definition and compare it with your earlier result.

Well fortunately, there are tables of the Laplace transforms of all kinds of special functions, including Bessel functions. One reference which has such a table is Abramowitz & Stegun, Handbook of Mathematical Functions, p. 1029:

http://people.math.sfu.ca/~cbm/aands/

If you don't trust such a reference, I suppose you could always derive the Laplace transform of your Bessel function from the definition and compare it with your earlier result.
Yah I'm supposed to derive it I'm sure.