I am actually writing a research paper on how to determine the approximate region of where a rolling ball in a roulette wheel will fall through a differential equation.
The differential equation is: $$\dfrac {dy}{dt}= -ay^2+b$$ where $(a)$ and $(b)$ are constants.
I found the answer to this differential equation in another research paper and the answer the research paper shows, but without any steps leading to it is: $$y(t)= -b\coth(c-abt)$$ where $c$ is the constant arising from the integration. I don't understand how this research paper managed to solve the nonlinear differential equation through the use of hyperbolic functions. Does anybody know how it has been done and is able to explain it please?
Let's make some of the algebra a little prettier by writing the ODE as
$\frac{dy}{dt} = -A^2 y^2 + B^2$
where $a = A^2$ and $b = b^2$. It doesn't matter too much if $a$ and $b$ are actually negative as long as you're comfortable with complex numbers.
Then this becomes a separable equation, meaning we can put all of the terms involving $y$ on one side and all of the terms involving $t$ on the other and then integrating, giving us:
$\begin{eqnarray} \int \frac{1}{B^2 - A^2 y^2} \frac{dy}{dt} & = & \int dt \\ \int \frac{dy}{B^2 - A^2 y^2} & = & t + C \\ \frac{1}{2B}\left( \int \frac{dy}{B - Ay} + \int \frac{dy}{B + Ay} \right) & = & t + C\end{eqnarray}$
using partial fraction decomposition. It then takes a little calculus and a little algebra, but you can then get to something involving logarithms which, if you know a suitable representation for $\coth^{-1}$, can be rearranged into the form you found.