\( x y ^ { \prime } - y = e ^ { y ^ { \prime } } \)

To solve the differential equation \( x y ' - y = e^{y '} \), one might first look for a way to separate the variables or use an appropriate substitution. Since the presence of \( y' \) is tricky, consider rewriting \( y' \) as \( \frac{dy}{dx} \) to clarify the dynamics of the variables involved, indicating that this equation involves both the function \( y \) and its derivative with respect to \( x \). A common mistake in approaching this type of equation is overlooking the implicit relationships between the variables. It's crucial to rearrange and isolate terms correctly before trying to integrate or differentiate further. Utilizing techniques like integrating factors or substituting variables can simplify the process. Don’t forget to check for any potential singularities in the solutions resulting from division by terms like \( x \) or \( y' \).