Bessel's Equation and substitutions

[lelandt50]

Homework Statement

Find a general solution in terms J_v of and Y_v . Indicate
whether you could also J_-v use instead of Y_v. Use the
indicated substitution. Show the details of your work.

9x²y''+9xy'+(36x⁴-16)y=0

Substitution (z=x²)

Homework Equations

All given in part 1.

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.

 

[HallsofIvy]

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

 

[lelandt50]

Thank you!

 

[DanGood]

i am also struggling with how I am going to compute