I have the following ODE and I am not sure how to solve it. I have manipulated it a bit, but am not sure if my approach is the correct thing to do.
$$ 0 = z^c-2(c-1)[1-z]y(z) + [2(1-z)-\mu]y'(z)-[\mu y']^2 $$
Dividing by $\mu^2$ to isolate the quadratic term
$$ 0 = \frac{z^c}{\mu^2}-\frac{2(c-1)}{\mu^2}[1-z]y(z) + \frac{1}{\mu^2}[2(1-z)-\mu]y'(z)-(y')^2 $$
Letting
$$ \alpha(z) = \frac{1}{\mu^2}[2(1-z)-\mu] $$
and
$$ \beta(z) = \frac{2(c-1)}{\mu^2}[1-z] $$
This becomes
$$ 0 = \frac{z^c}{\mu^2}-\beta y(z) + \alpha y'(z)-(y')^2 $$
or
$$ (y')^2 - \alpha y'(z) +\beta y(z) -\frac{z^c}{\mu^2}= 0 $$
However, I am not sure that it is acceptable to wrap the $1-z$ terms away into $\alpha$ and $\beta$.
If it is acceptable, then I will take the following square root of my expression
$$ y' = \pm \sqrt{\alpha y' -\beta y +\frac{z^c}{\mu}} $$
This is not separable, so I don't think I can integrate it directly. Additionally, I am concerned about the "hidden" $1-z$ terms wrapped into $\alpha$ and $\beta$.