Let $b\in \mathbb{R},\, \sigma>0, \, x \in \mathbb{R}$ be constants and $X$ the solution of the SDE $$ dX_t= b X_t dt + \sigma X_t dW_t , \; \; \; \; \; \; \; \; X_0 = x $$ Find $\mathbb{E} X_t^k$ for all natural numbers $k$ and $t \geq 0 $ fixed.
I have tried using Itò for $X^k$ but then realized I cannot even solve the questions for $k=1$
For completeness, applying Itô's lemma to $Y_t = log(X_t)$ as julien mentioned, or postulating a solution of the form $X_t = X_0 \exp(at + bW_t)$, one obtains: \begin{equation} X_t = X_0 \exp\left( \left( r-\frac{1}{2} \sigma^2 \right) t + \sigma W_t\right). \end{equation} Then, taking expectations, \begin{equation} \mathbb{E}[X_{t}^{k}] = X_0 \exp\left( \left( r-\frac{1}{2} \sigma^2 \right) kt \right) \mathbb{E}\left[ e^{k \sigma W_t} \right] = \exp\left( \left( r-\frac{1}{2} k\sigma^2 + \frac{1}{2} k^2 \sigma^2 \right) t \right), \end{equation} by the moment generating function of $k \sigma W_t \sim N(0,k^2 \sigma^2t) $.