Question

The general solution of the differential equation $\frac{dy}{dx}=e^{x+y}$, is
(a) e^x + e^−y = C
(b) e^x + e^y = C
(c) e^−x + e^y = C
(d) e^−x + e^−y = C

Answer 1

(a) e^x + e^−y = C

We have,
$$\begin{gathered} \frac{dy}{dx}=e^{x+y} \\ \Rightarrow \frac{dy}{dx}=e^x\times e^y \\ \Rightarrow e^{-y}dy=e^{x}dx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \int e^{-y}dy=\int e^{x}dx \\ \Rightarrow -e^{-y}=e^x+D \\ \Rightarrow e^x+e^{-y}=-D \\ \Rightarrow e^x+e^{-y}=C\left[\mathrm{Where},C=-D\right] \end{gathered}$$