PROBLEM Solve $$y'=a(x)y^2+b(x)$$
$a(x) = \frac{a_1}{m-M x} , b(x) = \frac{b_1-(m-M x)b_2}{m-M x} $ and $a_1,m,M,b_1,b_2 $ are constants
ADDITIONAL INFORMATION
I find on net it's probably Riccati Equation and they are computed by substitution. Any ideas on solving this equation?
$$y'=a(x)y^2+b(x)\tag 1$$ Equation $(1)$ is a Riccati ODE.
The change of function : $\quad y=-\frac{1}{a(x)}\frac{u'(x)}{u(x)}\quad$ transforms it into a linear second order ODE: $$a(x)u''-a'(x)u'+(a(x))^2b(x)u=0\tag 2$$ Since the analytical solutions of only a few linear second order ODEs are known on the form of combination of a finite number of elementary and special functions, there is no general method to express the solutions of equation $(2)$ on explicit form.
Thus there is no general method to express the solutions of equation $(1)$ on explicit form. This can be done only in some particular cases depending on the kind of functions $a(x)$ and $b(x)$.
Analytical solving is possible in the case of $a(x) = \frac{a_1}{m-M x} , b(x) = \frac{b_1-(m-M x)b_2}{m-M x} $
Compute $a'(x)$ and put it into equation $(2)$ as well as $b(x)$. This leads to an equation of Bessel kind. Eq.(6) in https://mathworld.wolfram.com/BesselDifferentialEquation.html