This is part of a physics problem - the solutions (as in 'answers') don't offer any explanation - they simply say:

I really can't see where (2) has come from. I've tried substituting in (3) into the original DE and get a horribly complicated expression which doesn't look anything like (2) - also I can't understand why 'p', the order of the Bessel's Equation, enters the DE (2).

I would suggest trying to solve your DE directly. x = 0 is a regular singular point, so you would look for a solution of the form

[tex] y = \sum_0^\infty a_nx^{n+r}[/tex]

Plug that in the equation using usual series methods. You should get an indicial equation of (r)(r-1)= 0. The roots differ by an integer so you can get one solution using r = 1. If you work it out you should get

[tex]J_1(x)= \sum_0^\infty \frac{(-1)^n}{n!\Gamma(n+2)}\left(\frac x 2\right)^{2n+1}[/tex]

Using this you should be able to calculate

[tex]\sqrt xJ_1(2\sqrt x)[/tex]

and get the series for y(x) above. You will have to review how to get the second solution when the indicial roots differ by an integer, but this should get you started.

Can I ask a stupid question though (my maths knowledge is very superficial): if we know that J_{1}(x) is a solution to a DE, isn't it always the case that Y_{1}(x) must also be part of the general solution (as it is linearly independent to to J_{1})?

Also can you offer any insight on why the original DE can be transformed(?) into this DE:

Hi. You know about Mathworld? Google it and then do a Bessel equation search and scroll down to the Bowman forumulation of Bessel's equation. The one with all the parameters. Now substitute your equation into that one. I get [itex]\alpha=1/2, n=1, \gamma=1/2, \beta=\pm 2[/itex] then substitute those into the solution given to arrive at