Use a stochastic representation result (feynman kac theorem) to solve the following boundary value problem :
$\frac{\partial V}{\partial t}+\mu x\frac{\partial V}{\partial x}+\frac{1}{2}\sigma^2x^2\frac{\partial^2 V}{\partial x^2}=0$
with $V(T,x)= \log(x^2)$
By Feynman-kac the solution should be $E[\log(X_T^2)|\mathscr{F}_t)]$ where we use the Ito process:$dX_t=\mu(X_t,t)dt+\sigma(X_t,t)dW_t$
and initial $X_t=x$.
Can I take $dX_t=\mu x dt+\sigma x dW_t$ here?
Which gives $X_T=x+\mu (T-t)+\sigma(W_T-W_t)$
I guess you got a typo a the start. Anyways, use $\mathrm{d} X_s = \mu X_s\mathrm{d}s + \sigma X_s\mathrm{d}W_s $ to get $$\log \frac{X_T}{X_t} = \left( \mu - \frac{1}{2}\sigma^2\right)(T-t) + \sigma (W_T - W_t)$$ Then $$ V(t,x) = \mathbb{E}[2\log X_T | \mathcal{F}_t] = 2\log x + (2\mu - \sigma^2)(T-t) $$