Trying to solve the Poschl-Teller potential

haimfeld

Trying to solve the Poschl-Teller potential (quantum mechanics)

1. The problem statement, all variables and given/known data
I have a superpotential that give me 2 of the Poschl-Teller potentials.
The superpotential is:
W(x)=-b*cot(x)
The Poschl-Teller potentials are:
J(x,b)=b(b-1)/(sin(x))^2-b^2
K(x,b)=b(b+1)/(sin(x))^2-b^2
2. Relevant equations
Schrodinger equation: Hψ=Eψ
3. The attempt at a solution
I placed the potential in Schrodinger equation (neglecting the existence of constants) and received the following equations:
ψ''(x)+(2E-J)ψ(x)=0
ψ''(x)+(2E-K)ψ(x)=0
I do not know what the next step I should do
I know I need to get to Legendre polynomial but I don't know how...
Can anyone show me how to do it?

Thaakisfox

Put [itex]u=\cos x [/itex]. Rewrite the DE in the u variable....