Bessel's equation and second order DE

Hi all!

I've hit a wall on the question I'm working on. Can someone look over my work and see where I may have possible gone wrong?

The question is:

Show that if is a solution of Bessel's equation of order p, then, is a solution of .

Here is my working.

Considering that is a solution, then:

and

Subing into ,

+ .

I simplify to find:

As , that means .

Making the substitution of

I get the following:

.

That equation is soo close to Bessel's equation. I go through my working and I can't find why I have those b's are there.

Thank you very much for your time.