Solve (n^2+y^2)dy+2nydx=0

$$( n ^ { 2 } + y ^ { 2 } ) d y + 2 n y d x = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&n=\sqrt{x^{2}-y^{2}}-x\text{ or }n=-\sqrt{x^{2}-y^{2}}-x\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply n^{2}+y^{2} by d.

$$\left(n^{2}d+y^{2}d\right)y+2nydx=0$$

Use the distributive property to multiply n^{2}d+y^{2}d by y.

$$n^{2}dy+dy^{3}+2nydx=0$$

Combine all terms containing d.

$$\left(n^{2}y+y^{3}+2nyx\right)d=0$$

The equation is in standard form.

$$\left(2nxy+y^{3}+yn^{2}\right)d=0$$

Divide 0 by n^{2}y+y^{3}+2nyx.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(n=-\sqrt{\left(x-y\right)\left(x+y\right)}-x\text{ and }|y|\leq |x|\text{ and }y\leq -x\text{ and }y\geq x\right)\text{ or }\left(n=-\sqrt{\left(x-y\right)\left(x+y\right)}-x\text{ and }|y|\leq |x|\text{ and }y\leq x\text{ and }y\geq -x\right)\text{ or }\left(n=-x\text{ and }|y|=|x|\right)\text{ or }\left(n=\sqrt{\left(x-y\right)\left(x+y\right)}-x\text{ and }|y|\leq |x|\text{ and }y\leq -x\text{ and }y\geq x\right)\text{ or }\left(n=\sqrt{\left(x-y\right)\left(x+y\right)}-x\text{ and }|y|\leq |x|\text{ and }y\leq x\text{ and }y\geq -x\right)\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply n^{2}+y^{2} by d.

$$\left(n^{2}d+y^{2}d\right)y+2nydx=0$$

Use the distributive property to multiply n^{2}d+y^{2}d by y.

$$n^{2}dy+dy^{3}+2nydx=0$$

Combine all terms containing d.

$$\left(n^{2}y+y^{3}+2nyx\right)d=0$$

The equation is in standard form.

$$\left(2nxy+y^{3}+yn^{2}\right)d=0$$

Divide 0 by n^{2}y+y^{3}+2nyx.

$$d=0$$

Solve for n (complex solution)

$$\left\{\begin{matrix}\\n=\sqrt{x^{2}-y^{2}}-x\text{; }n=-\sqrt{x^{2}-y^{2}}-x\text{, }&\text{unconditionally}\\n\in \mathrm{C}\text{, }&y=0\text{ or }d=0\end{matrix}\right.$$

Solve for n

$$\left\{\begin{matrix}n=\sqrt{x^{2}-y^{2}}-x\text{; }n=-\sqrt{x^{2}-y^{2}}-x\text{, }&|y|\leq |x|\\n\in \mathrm{R}\text{, }&y=0\text{ or }d=0\end{matrix}\right.$$