# Solve differential equation \frac{dy}{dx}= \frac{x-y-1}{x+y+3}

Question
Solve differential equation $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{{x}-{y}-{1}}}{{{x}+{y}+{3}}}}$$

2021-02-19
On substituting
$$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={y}^{{t}}$$
On rewriting in the form of an exact differential equation
$$\displaystyle-{x}+{y}+{1}+{\left({x}+{y}+{3}\right)}{y}^{{t}}={0}$$
To verify the following condition
$$\displaystyle{\frac{{\partial{M}{\left({x},{y}\right)}}}{{\partial{y}}}}={\frac{{\partial{N}{\left({x},{y}\right)}}}{{\partial{x}}}}$$
Thus the above condition is true
$$\displaystyle{3}{y}+{x}{y}+{\frac{{{y}^{{2}}}}{{{2}}}}-{\frac{{{x}^{{2}}}}{{{2}}}}+{x}+{c}_{{1}}={c}_{{2}}$$
$$\displaystyle{3}{y}+{x}{y}+{\frac{{{y}^{{2}}}}{{{2}}}}-{\frac{{{x}^{{2}}}}{{{2}}}}+{x}={c}_{{1}}$$
PSKy= -3-x+\sqrt{2x^2+4x+c_1+9}PSK
$$\displaystyle{y}=-{x}-\sqrt{{{2}{x}^{{2}}+{4}{x}+{c}_{{1}}+{9}}}-{3}$$

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