Solve the differential equation : $\frac{d y}{d x}=e^{x+y}$
Solve the differential equation : $\frac{d y}{d x}=e^{x+y}$
- $e^x+e^y=c$
- $e^x-e^y=c$
- $e^x+e^{-y}=c$
- $e^x-e^{-y}=c$
Solution
$
\text { } \begin{aligned}
\frac{d y}{d x} & =e^{x+y} \\
\frac{d y}{d x} & =e^x \cdot e^y \Rightarrow \frac{d y}{e^y}=e^x d x \\
e^{-y} d y & =e^x d x
\end{aligned}
$
Integrating both sides,
$
\begin{aligned}
& \int e^{-y} d y=\int e^x d x \\
& \Rightarrow \quad \frac{e^{-y}}{-1}=e^x+c \Rightarrow e^x+e^{-y}+c=0 \\
& \Rightarrow \quad e^x+e^{-y}=-c \Rightarrow e^x+e^{-y}=c \\
&
\end{aligned}
$
when $c=-c$
Asked in: AP EAMCET 2021 (24 Aug Shift 1)
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