Solve xdx+ydy+(x^2+y^2)dy=0

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

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&y=-1\text{ or }x=iy\text{ or }x=-iy\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply x and x to get x^{2}.

$$x^{2}d+ydy+\left(x^{2}+y^{2}\right)dy=0$$

Multiply y and y to get y^{2}.

$$x^{2}d+y^{2}d+\left(x^{2}+y^{2}\right)dy=0$$

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

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

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

$$x^{2}d+y^{2}d+x^{2}dy+dy^{3}=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide 0 by x^{2}+y^{2}+x^{2}y+y^{3}.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(y=0\text{ and }x=0\right)\text{ or }y=-1\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply x and x to get x^{2}.

$$x^{2}d+ydy+\left(x^{2}+y^{2}\right)dy=0$$

Multiply y and y to get y^{2}.

$$x^{2}d+y^{2}d+\left(x^{2}+y^{2}\right)dy=0$$

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

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

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

$$x^{2}d+y^{2}d+x^{2}dy+dy^{3}=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide 0 by x^{2}+y^{2}+x^{2}y+y^{3}.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-iy\text{; }x=iy\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\text{ or }y=-1\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=0\text{, }&y=0\\x\in \mathrm{R}\text{, }&d=0\text{ or }y=-1\end{matrix}\right.$$