Let bessel's equation be written as
$ z^2$$\frac{d^2w}{dz^2}$ + $z$$\frac{dw}{dz}$ + $(z^2 -p^2)w$ = $0$
and show that the change of variables defined by $z = ax^b$ and $w = yx^c$ (where a,b and c are constants) transforms it into
$ x^2$$\frac{d^2y}{dx^2}$ + $(2c+1)x\frac {dy}{dx}$ + $[a^2b^2(x)^{2b} + (c^2 - p^2b^2)]y$ = $0$
Write the general solution of this equation in terms of Bessels function.
I also don't understand why we use this transformation. Any help would be apreciated.
It is because you are invited to chose $b$ and $c$ such that the constant $c^2 - p^2b^2$ is equal to $0$. In this way you obtain a simpler differential equation...
It is because the solution of the FIRST equation can be written as c1*F(z)+c2*G(z), where c1, c2 are computed based on the boundary conditions and F, G are Bessel functions.
Therefore, any equation that follows the SECOND equation pattern can be tranformed into an equivalent first equation, and thus be easily solved.
