Step 1 :Given the differential equation \(y^{\prime}=\frac{e^{2 x}}{\ln y}\).
Step 2 :This is a first order differential equation. It can be solved by separating variables and integrating.
Step 3 :Rewrite the equation as \(y' \ln y = e^{2x}\).
Step 4 :Integrate both sides of the equation. The integral of the left side is \(y \ln y - y\) and the integral of the right side is \(\frac{e^{2x}}{2}\).
Step 5 :The general solution should be in the form of \(y \ln y - y = \frac{e^{2x}}{2} + C\), where \(C\) is the constant of integration.
Step 6 :Comparing the form of the solution obtained with the options given in the question, the general solution for the differential equation is \(\boxed{y \ln y - y = \frac{e^{2x}}{2} + C}\), which corresponds to option D.