Solution to sde with specfic mean

Question

Goal: I'm attempting to work backwards to recover an SDE as follows:

Example: $e^{\mu t}$ is the mean of the geometric Brownian Motion, which solves the SDE: \begin{equation} dS_t = \mu S_t dt + \sigma dW_t \mbox{; } (t \in (0,\infty)). \end{equation}

Question: What SDE has a solution which has mean $t^ke^{\mu t}$.

Answer 1

$$ dS_t = \Big(\mu + \frac{1}{t}\Big) S_t\,dt + \sigma dW_t $$ for example. Of course this comes with a restriction to $t > 0$.