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

Given, $\frac{dy}{dx}=e^{x+y}\Rightarrow \frac{dy}{dx}=e^x{e}^y$
On separating the variables, we get $e^{-y}dy=e^{x}dx$
On integrating both sides, we get $\int e^{-y}dy=\int e^{x}dx\Rightarrow \frac{e^{-y}}{-1}=e^x+A$
$\Rightarrow e^x+e^{-y}=-A\Rightarrow e^x+e^{-y}=C,whereC=-A$
Hence, (a) is the correct option.