Jacobi Differential Equation

$$(1-x^2)y''+[\beta-\alpha-(\alpha+\beta+2)x]y'+n(n+\alpha+\beta+1)y=0$$ (1)

or

$${d\over dx}[(1-x)^{\alpha+1}(1+x)^{\beta+1}y']+n(n+\alpha+\beta+1)(1-x)^\alpha(1+x)^\beta y=0.$$ (2)

The solutions are Jacobi Polynomials. They can be transformed to

$${d^2u\over dx^2}+\left[{{1\over 4}{1-\alpha^2\over(1-x)^2}+{... ...extstyle{1\over 2}}(\alpha+1)(\beta+1)\over 1-x^2}}\right]u=0,$$ (3)

where

$$u=u(x)=(1-x)^{(\alpha+1)/2}(1+x)^{(\beta+1)/2}P_n^{(\alpha,\beta)}(x),$$ (4)

and

$${d^2u\over d\theta^2}+\left[{{{\textstyle{1\over 4}}-\alpha^... ...theta)}+\left({n+{\alpha+\beta+1\over 2}}\right)^2}\right]u=0,$$ (5)

where

$$u=u(\theta)=\sin^{\alpha+1/2}({\textstyle{1\over 2}}\theta)\... ...\textstyle{1\over 2}}\theta) P_n^{(\alpha,\beta)}(\cos\theta).$$ (6)