Change of variables in Seocnd order ODES

[phil ess]

I am looking through my course notes for mathematical physics, in preparation for the exam, and I've run into a concept that I can't figure out. It comes up first when talking about the modified bessel's equation

(x^2)y''+(x)y'-(x^2+p^2)y=0

And supposedly this can be transformed into bessel's euqation by using the substitution t=ix, but I can't figure out how this works. This is just the specific example that this issue came up in; I don't understand generally how to perform a change of variables for second order equations.

I start with dt=idx and -t^2=x^2, but how do I deal with y'' and y'?

If anyone can explain how change of variables works to me it would be greatly appreciated!

 

[JJacquelin]

dt = i dx
dx = -i dt

dy/dx = dy/(-i dt) = i dy/dt

d²y/dx² = d(dy/dx)/dx = d(i dy/dt) / (-i dt) = - d²y/dt²