# Bessel's Equation and substitutions

1. Nov 9, 2013

### lelandt50

1. The problem statement, all variables and given/known data
Find a general solution in terms Jv of and Yv . Indicate
whether you could also J-v use instead of Yv. Use the
indicated substitution. Show the details of your work.

9x2y''+9xy'+(36x4-16)y=0

Substitution (z=x2)

2. Relevant equations

All given in part 1.

3. The attempt at a solution

Given z=x2, $\frac{dz}{dx}$=2x
Therefore $\frac{dy}{dx}$=$\frac{dy}{dz}$*$\frac{dz}{dx}$=2x*$\frac{dy}{dz}$

But I need the second derivative of y with respect to x to make the substitution, this is where I run into trouble. Using the chain rule, I get this:
$\frac{d^{2}y}{dx^{2}}$=2*$\frac{dy}{dz}$+$\frac{d}{dx}$($\frac{dy}{dz}$)*2x

I have no clue how to compute $\frac{d}{dx}$($\frac{dy}{dz}$)
Any help would be appreciated.

2. Nov 9, 2013

### HallsofIvy

Staff Emeritus
For any function $\phi$, $\frac{d\phi}{dx}= \frac{d\phi}{dz}\frac{dz}{dx}$. In particular, if $\phi= \frac{dy}{dz}$ then $\frac{d}{dx}\left(\frac{dy}{dz}\right)= \frac{d^2y}{dz^2}\left(\frac{dz}{dx}\right)$

3. Nov 9, 2013

### lelandt50

Thank you!

4. Feb 17, 2016

### DanGood

i am also struggling with how im going to compute