The general form of the Jacobi ODE is given by the following:

Jacobi_ode := diff(y(x),x,x)*x*(1-x) = (g-(a+1)*x)*diff(y(x),x)+n*(a+n)*y(x);

$\mathrm{Jacobi\_ode}≔\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)x\left(1-x\right)=\left(g-\left(a+1\right)x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+n\left(a+n\right)y\left(x\right)$

where n is an integer. See Iyanaga and Kawada, "Encyclopedic Dictionary of Mathematics", p. 1480.

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{odeadvisor}\right)$

$\left[\mathrm{odeadvisor}\right]$

$\mathrm{odeadvisor}\left(\mathrm{Jacobi\_ode}\right)$

$\left[\mathrm{\_Jacobi}\right]$

$\mathrm{dsolve}\left(\mathrm{Jacobi\_ode}\right)$

$y\left(x\right)=\mathrm{c\_\_1}\mathrm{hypergeom}\left(\left[-1-\frac{a}{2}-\frac{\sqrt{a^2+\left(-4n+4\right)a-4n^2+4}}{2}\,-1-\frac{a}{2}+\frac{\sqrt{a^2+\left(-4n+4\right)a-4n^2+4}}{2}\right]\,\left[-g\right]\,x\right)+\mathrm{c\_\_2}{x}^{1+g}\mathrm{hypergeom}\left(\left[-\frac{a}{2}-\frac{\sqrt{a^2+\left(-4n+4\right)a-4n^2+4}}{2}+g\,-\frac{a}{2}+\frac{\sqrt{a^2+\left(-4n+4\right)a-4n^2+4}}{2}+g\right]\,\left[2+g\right]\,x\right)$