I am interested in the solution to the following 1-parameter family of 2nd order ODE: $$(\lambda (xH' - H)^3 + 4(H')^2 (1-H') x^2)H'' - 4 H'(1-H')(xH'-H)(x+H')=0,$$
with $H(0) = 1$ and $H(1) = 0$. $\lambda < 0$ is the parameter.
I am interested in either analytic solution or proof that no analytic solutions exists. Uniqueness of solutions is also appreciated. Thanks!