The general solution of the differential equation d y d x =e^ x+y… - Vidyadip

MCQ

The general solution of the differential equation $\frac{d y}{d x}=e^{x+y}$ is

A
$e^{-x}+e^{y}=C$
B
$e^{x}+e^{y}=C$
✓
$ e^{x}+e^{-y}=C$
D
$e^{-x}+e^{-y}=C$

✓

Answer

Correct option: C.

$ e^{x}+e^{-y}=C$

c
$\frac{d y}{d x}=e^{x+y}=e^{x} \cdot e^{y}$

$\Rightarrow \frac{d y}{e^{y}}=e^{x} d x$

$\Rightarrow \mathrm{e}^{-\mathrm{y}} \mathrm{dy}=\mathrm{e}^{\mathrm{x}} \mathrm{d} \mathrm{x}$

Intergrating both sides, we get:

$\int e^{-y} d y=\int e^{x} d x$

$\Rightarrow-e^{-y}=e^{x}+k$

$\Rightarrow \mathrm{e}^{\mathrm{x}}+\mathrm{e}^{-\mathrm{y}}=-\mathrm{k}$

$\Rightarrow e^{x}+e^{-y}=c \quad(c=-k)$

Hence, the correct answer is $\mathrm{C}.$

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