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

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

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

Jeffrey Jordon