I have an ODE of the form

$C_1 d_y u + C_2 (d_y u)^n = C_3 y + C_4$

with boundary conditions $u(0) = 0, d_y u(L) = 0$, where $C_1 \to C_4$ are known constants, and where $0 < n \leq 1$ is a real number.

Special cases:

If $C_2 = 0$, this is a simple linear ODE and is straightforward to solve.

If $C_1 = 0$, I can take the $\sqrt[n]{}$ of both sides and proceed to solve the nonlinear first order ODE.

Question: Is there then an appropriate method for solving the general case ($C_1 \neq 0, C_2 \neq 0$)?