Sturm Liouville ODE Bessel Functions

[jborcher]

Homework Statement

x d²y(x)/dx² + dy(x)/dx + 1/4 y(x)

Show that the solution can be obtained in terms of Bessel functions J₀.

Homework Equations

Hint: set u = x^a where a is not necessarily an integer. Judiciously select a to get y(u).

The Attempt at a Solution

I tried just straight pluggin in x=u^1/a and ended up with the following form for the diff eq:

u² d²y(u)/du² + (1-a)/a u^1-a-1 dy(u)/du + (1-a)/4a² y(u) = 0

I've hit a wall here, this doesn't match the Bessel Equation (though I am pretty sure it is not supposed to). I am unsure how to select a in order to get a solution with J₀.

I tried another approach where I followed the various differentiation rules for Bessel functions and obtained the following:

-x J₀(x) + 1/4 J₀(x) = 0

Again I have hit a wall and am not sure how I should proceed.

 

[jackmell]

Homework Statement

x d²y(x)/dx² + dy(x)/dx + 1/4 y(x)

Show that the solution can be obtained in terms of Bessel functions J₀.

Homework Equations

Hint: set u = x^a where a is not necessarily an integer. Judiciously select a to get y(u).

The Attempt at a Solution

I tried just straight pluggin in x=u^1/a and ended up with the following form for the diff eq:

u² d²y(u)/du² + (1-a)/a u^1-a-1 dy(u)/du + (1-a)/4a² y(u) = 0

I've hit a wall here, this doesn't match the Bessel Equation (though I am pretty sure it is not supposed to). I am unsure how to select a in order to get a solution with J₀.

I tried another approach where I followed the various differentiation rules for Bessel functions and obtained the following:

-x J₀(x) + 1/4 J₀(x) = 0

Again I have hit a wall and am not sure how I should proceed.

Best way to learn this in my opinion is to just get the answer first and then work towards it so that you have practice for the next one. So the Bessel DE is:

$$v^2 \frac{d^2y}{dv^2}+v \frac{dy}{dv}+(v^2+a^2)y=0$$

and Mathematica gives the solution in terms of $J(0,\sqrt{x})$. So then let's just let $u=x^{1/2}$. No that's not cheating. Do you want to just eat fish or learn how to fish? Ok, can we just get the solution with that substitution and then if you want to, solve it using $u=x^{n/m}$ to see why 1/2 works for more practice.

You can do all those chained-derivatives right?

$$\frac{dy}{du}=2u\frac{dy}{dx}$$

$$\frac{d^2y}{du^2}=\frac{1}{u}\frac{dy}{du}+4u^2 \frac{d^2 y}{dx^2}$$

and you can finish it to arrive at the Bessel form of the DE.