I am stuck in trying to solve this equation \begin{align} d X_t = - b^2 X_t (1 - X_t)^2 dt + b \sqrt{1 - X_t^2} dW_t \end{align}
Here, $b$ is a constant. I am trying to apply my usual methods for solving SDEs, but nothing is working out. E.g. I tried applying Ito's formula to $h(X_t)$ and choosing $h$ so that the $dW_t$ term is constant, but it didn't work out. I also tried setting $X_t = h(t, W_t)$, applying Ito's formula and choosing $h$ to match the terms in the SDE. That also did not work out. What is the correct method here? N.B. I have the solution and can provide it if necessary. But I really would like to know the method that is used. Many thanks.
Edit: Here is another thing that I tried. Once again, I applied Ito's formula to $X_t = h(t, W_t)$. Then, in the equation for $d[h(X_t)]$, I tried to make the $dt$ term equal to a constant (say $1$ or $b^2$). When I did this, I got a second order ODE. I reduce the order of the ODE and tried to solve it using the integrating factor method. But the integral in the integrating factor doesn't integrate exactly, and so I know that I am going down the wrong path. Please could somebody steer me in the right direction? Many thanks.
In your system of PDEs in the term: $ \frac{1}{2} b^{2} \frac{\partial^{2}h}{\partial w^{2}} (1-h^{2}) $
Don't you have an extra $b^{2}$ there?