How to solve the following SDEs?

Question

Consider the SDEs

$dx_t = x_t \log x_t\text{d}t + x_t\text{d}W_t$

$dx_t = x_t^2 \text{d}t + x_t\text{d}W_t$

How do you solve something like these? I tried $x_t=F(W_t,t)$, but could not get a solution.

Answer 1

For the first equation, note that \begin{align*} d\ln x_t = (\ln x_t -\frac{1}{2}) dt + dW_t, \end{align*} and \begin{align*} d\left(e^{-t}\ln x_t \right) &=-\frac{1}{2}e^{-t}dt + e^{-t} dW_t. \end{align*} Then $x_t$ can be expressed as an integral form.