1. The problem is to find $\frac{dy}{dx}$ given the differential equation $\frac{dy}{dx} = x \ln y$. 2. This is a separable differential equation. We can rewrite it as $\frac{dy}{dx} = x \ln y$. 3. Separate variables: $\frac{dy}{\ln y} = x dx$. 4. To integrate the left side, use substitution: let $u = \ln y$, then $du = \frac{1}{y} dy$ or $dy = y du = e^u du$. 5. Substitute into the integral: $\int \frac{dy}{\ln y} = \int \frac{e^u du}{u}$. 6. The integral $\int \frac{e^u}{u} du$ does not have an elementary antiderivative, so the solution involves the Exponential Integral function $\text{Ei}(u)$. 7. Integrate the right side: $\int x dx = \frac{x^2}{2} + C$. 8. Therefore, the implicit solution is $\text{Ei}(\ln y) = \frac{x^2}{2} + C$. 9. This expresses $y$ implicitly in terms of $x$ using the Exponential Integral function. Final answer: $$\text{Ei}(\ln y) = \frac{x^2}{2} + C$$