Q.Solve $\displaystyle\frac{dy}{dx} = e^{x - y}$.

Split $e^{x-y}=e^x e^{-y}$ so the variables separate.

Write $e^{x-y} = e^{x}\,e^{-y}$, so the equation becomes

Separate the variables (multiply both sides by $e^{y}\,dx$):

Integrate both sides:

Verify (differentiate): from $e^{y} = e^{x}+c$, differentiate with respect to $x$: $e^{y}\dfrac{dy}{dx} = e^{x}$, so $\dfrac{dy}{dx} = e^{x}\,e^{-y} = e^{x-y}$ — the original equation. Correct.

Use the law $e^{x-y}=e^x\cdot e^{-y}$ to split the exponential so the $y$-part can move to the left — do not try to integrate $e^{x-y}$ as a single block.