Question

The general solution of the differential equation $\frac{dy}{dx}=e^{x-y}$ is _______________.

Answer 1

Given: $\frac{dy}{dx}=e^{x-y}$

$$\begin{gathered} \frac{dy}{dx}=e^{x-y} \\ \Rightarrow \frac{dy}{dx}=e^x{e}^{-y} \\ \Rightarrow \frac{dy}{dx}=\frac{e^x}{e^y} \\ \Rightarrow e^{y}dy=e^{x}dx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \Rightarrow \int e^{y}dy=\int e^{x}dx \\ \Rightarrow e^y=e^x+C,\mathrm{where}C\mathrm{is}\mathrm{any}\mathrm{constant} \end{gathered}$$

Hence, the general solution of the differential equation $\frac{dy}{dx}=e^{x-y}$ is $\underline{e^y=e^x+C}$.