Find the general solution of the differential equation d y d x =e… - Vidyadip

Question

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

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Answer

Given differential equation is
$\frac{d y}{d x}=e^{x+y}$
$\Rightarrow \quad \frac{d y}{d x}=e^x e^y$
$\Rightarrow \quad \frac{d y}{e^y}=\left(e^x\right) d x$
$\Rightarrow \quad\left(e^{-y}\right) d y=\left(e^x\right) d x$
Integrating both sides, we get
$\int\left(e^{-y}\right) d y=\int\left(e^x\right) d x$
$\Rightarrow \quad-e^{-y}=e^x+c^{\prime}$
$\Rightarrow \quad e^{-y}=-e^x+c[$ where $c=-\dot{c}$

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