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## Homework Statement

I am trying to solve this

\begin{align}

d X_t = - b^2 X_t (1 - X_t)^2 dt + b \sqrt{1 - X_t^2} dW_t

\end{align}

where $b$ is a constant.

Note that I have the answer here and can provide it if necessary. But I want to know how one would come up with it.

## Homework Equations

## The Attempt at a Solution

I do not know many general methods for finding analytical expressions for the solutions of SDEs. The main method I know is this. Apply Ito's formula to $h(X_t)$ and where $d X_t$ appears, of course use the SDE given. Then you hope that you can choose h such that both the integrands are equal to a constant, or don't depend on $X_t$. Then you'd have an exact expression for $h(X_t) - h(X_0)$ and you would be able to proceed from there. That method did not seem to work here. Another method that I tried is to hope that we can get $X_t = f(W_t)$. We apply Ito's formula to $f(W_t)$ and match the integrands to those in $dX_t$, replace $X_t$ everywhere with $f(W_t)$. But I can't seem to match both integrands. I matched the $dW_t$ term to get $f(W_t) = sin(b W_t)$. But this will not match with the $dt$ integrand. So both of the methods that I know did not work. How can I solve it? Many thanks.