Solve the following differential equations: \frac{dy}{dx}=\frac{x-e^{-x}}{y+e^{y}}

mattgondek4 2021-02-27 Answered
Solve the following differential equations:
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{{x}-{e}^{{-{x}}}}}{{{y}+{e}^{{{y}}}}}}\)

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Expert Answer

Obiajulu
Answered 2021-03-01 Author has 24797 answers
Step 1
Given
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{{x}-{e}^{{-{x}}}}}{{{y}+{e}^{{{y}}}}}}\)
To Find- The value of the above differential equations.
Step 2
Explanation- Rewrite the given expression,
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{{x}-{e}^{{-{x}}}}}{{{y}+{e}^{{{y}}}}}}\)
Simplifying the above expression and integrating both sides, we get,
\(\displaystyle\int{\left({y}+{e}^{{{y}}}\right)}{\left.{d}{y}\right.}=\int{\left({x}-{e}^{{-{x}}}\right)}{\left.{d}{x}\right.}\)
\(\displaystyle{\frac{{{y}^{{{2}}}}}{{{2}}}}={e}^{{{y}}}={\frac{{{x}^{{{2}}}}}{{{2}}}}+{e}^{{-{x}}}+{C}\)
\(\displaystyle{\frac{{{y}^{{{2}}}}}{{{2}}}}-{\frac{{{x}^{{{2}}}}}{{{2}}}}+{e}^{{{y}}}-{e}^{{-{x}}}={C}\)
Answer- Hence, the solution of the differential equation \(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{{x}-{e}^{{-{x}}}}}{{{y}+{e}^{{{y}}}}}}\) is
\(\displaystyle{\frac{{{y}^{{{2}}}}}{{{2}}}}-{\frac{{{x}^{{{2}}}}}{{{2}}}}+{e}^{{{y}}}-{e}^{{-{x}}}={C}\).
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Answered 2021-11-20 Author has 11052 answers

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