Get transfer function of a nonlinear diff. equation

Question

I have this equation:

$$\frac{\partial v}{\partial t} = -g + c\left(u(t) - v(t)\right)^2$$

g and c are constants. u(t) is my input and v(t) is my output. I need to reach the transfer function $\frac{V(s)}{U(s)}$ at Laplace domain. In the way the equation is, I think I just can't do it, so I have to linearize it. I read that I can use Taylor series, but I'm getting stucked... How can I get this transfer function?

Some info:

$v(0) = 0 \\ if \ \ u(t) = k, v(t) = 0$

Answer 1

HINT

$$ v'(t)=-g+c u^2(t)-2 c u(t) v (t)+c v^2(t) $$ This is a Riccati's Equation, of the form $$ v'(t) = q_0(t) + q_1(t) \, v(t) + q_2(t) \, v^2(t) $$ where $q_0(t)= -g+c u^2(t)\neq 0$, $q_1(t)=-2 c u(t)$ and $q_2(t)=c\neq 0$.