- #1

Crush1986

- 207

- 10

## Homework Statement

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

## Homework Equations

[tex] -(d/ds)L{f(x)} [/tex]

## 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...

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

This gives me [tex] -2sY(s)+1+sY(s)-1=0 [/tex]

=> [tex] -sY(s)=0 [/tex]

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.