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AP EAPCET 24-Aug-2021 Shift 1 Paper

Section: Mathematics

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Question : 80 of 160

Marks: +1, -0

Solve the differential equation:

\frac{dy}{dx}=e^{x+y}

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

Solution:

\frac{dy}{dx}=e^{x+y}

\frac{dy}{dx}=e^{x} \cdot e^{y}

\Rightarrow \frac{dy}{e^{y}}=e^{x} dx

e^{-y} dy=e^{x} dx

Integrating both sides,

\int e^{-y} dy=\int e^{x} dx

\Rightarrow \frac{e^{-y}}{-1}=e^{x}+c

\Rightarrow e^{x}+e^{-y}+c=0

\Rightarrow e^{x}+e^{-y}=-c

\Rightarrow e^{x}+e^{-y}=c

when

c=-c

\Rightarrow v_{av}=\frac{2 v_{1} v_{2}}{v_{1}+v_{2}}

By substituting the values, we get

v_{av}=\frac{2 \times 18 \times 36}{18+36}

=\frac{2 \times 36 \times 18}{54}=\frac{72}{3} \text{m/s}

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