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