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

\(\displaystyle{\frac{{{y}^{{{2}}}}}{{{2}}}}-{\frac{{{x}^{{{2}}}}}{{{2}}}}+{e}^{{{y}}}-{e}^{{-{x}}}={C}\).

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

\(\displaystyle{\frac{{{y}^{{{2}}}}}{{{2}}}}-{\frac{{{x}^{{{2}}}}}{{{2}}}}+{e}^{{{y}}}-{e}^{{-{x}}}={C}\).