\( 3\frac{dy}{dx}=\frac{y+\frac{x}{y^{2}}}{3y^{2}-x} \)

To solve the given differential equation \( 3\frac{dy}{dx}=\frac{y+\frac{x}{y^{2}}}{3y^{2}-x} \), we start by rearranging it into a more manageable form. Cross-multiplying allows us to isolate \( dy \) and \( dx \), giving us: \[ 3(3y^2 - x) dy = (y + \frac{x}{y^2}) dx. \] Next, we can identify potential substitution or separation of variables techniques to simplify our work. Often in such equations, testing for any homogeneous characteristics or integrating factors can yield fruitful results. A common mistake to avoid is neglecting to check the conditions of the variables and ensuring the solution is valid in the domain of the original equation. Keeping track of the dependencies between \( x \) and \( y \) is key during integration.