Solve (y+x)dy-(y-x)dx=0

$$( y + x ) d y - ( y - x ) d x = 0$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

Use the distributive property to multiply y+x by d.

$$\left(yd+xd\right)y-\left(y-x\right)dx=0$$

Use the distributive property to multiply yd+xd by y.

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

Use the distributive property to multiply y-x by d.

$$dy^{2}+xdy-\left(yd-xd\right)x=0$$

Use the distributive property to multiply yd-xd by x.

$$dy^{2}+xdy-\left(ydx-dx^{2}\right)=0$$

To find the opposite of ydx-dx^{2}, find the opposite of each term.

$$dy^{2}+xdy-ydx+dx^{2}=0$$

Combine xdy and -ydx to get 0.

$$dy^{2}+dx^{2}=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for d

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

Steps for Solving Linear Equation

Use the distributive property to multiply y+x by d.

$$\left(yd+xd\right)y-\left(y-x\right)dx=0$$

Use the distributive property to multiply yd+xd by y.

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

Use the distributive property to multiply y-x by d.

$$dy^{2}+xdy-\left(yd-xd\right)x=0$$

Use the distributive property to multiply yd-xd by x.

$$dy^{2}+xdy-\left(ydx-dx^{2}\right)=0$$

To find the opposite of ydx-dx^{2}, find the opposite of each term.

$$dy^{2}+xdy-ydx+dx^{2}=0$$

Combine xdy and -ydx to get 0.

$$dy^{2}+dx^{2}=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for x (complex solution)

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

Solve for x

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