# Equation involving difference of beta CDFs

Consider the expression $$I_{p}(\alpha,\beta+1) - I_{p}(\alpha+1,\beta) = c$$ where $$I_p(a,b)$$ is the regularized incomplete beta function.

Question: Given $$\alpha,\beta,$$ and $$c>0$$, what is $$p$$?

Attempt: I need to invert the CDF's, but the problem is the arguments of the CDFs are not the same. I'm not sure how to proceed.

$$I_p(\alpha,\beta+1)=I_p(\alpha,\beta)+\frac{p^{\alpha}(1-p)^{\beta}}{\beta\,B(\alpha,\beta)}$$
$$I_p(\alpha+1,\beta)=I_p(\alpha,\beta)-\frac{p^{\alpha}(1-p)^{\beta}}{\alpha\,B(\alpha,\beta)}$$
$$p^{\alpha}\,(1-p)^{\beta}=c\,(\frac{1}{\alpha}+\frac{1}{\beta})\,B(\alpha,\beta)\:$$
From here there is no general solution in a closed form $$\forall\,\alpha$$,$$\beta$$ but last equation is very easy to solve numerically