This is a homework exercise. I am asked to show that the stochastic process $X_t = B_t^2$ with $B_t$ being a standard brownian motion, is a weak solution to
$$ dX_t = dt + 2\sqrt{|X|}d\tilde{B}, \quad X_0 = 0 $$ Where $\tilde{B}$ is Another standard brownian motion.
My attempt so far has been to try and use the Itô formula to solve the equation as we have trained in class.
My attempt at a solution was to follow a similar schema as when I found a solution to the same SDE but with $\tilde{B} = B_t, \quad X_0;= x_0 $. The change being that I set $g(t,x) = x^2$ such that $g' = 2x, g'' = 2$ and substitute into the equation $$ dg(B_t) = g(0) + g'dB_t (?) + \frac{1}{2}g'' dt $$
For the other problem this startegy was succesful when choosing $g(t,x) = 2\sqrt{x}$, I was able to follow the literature and derive the correct solution.
Nevertheless I feel very stuck in showing that $X_t = B_t^2$ is indeed a weak solution and need help. How do I show that it is a weak solution?