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

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

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

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

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

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

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

$$dx^{2}+ydx-\left(xd-yd\right)y=0$$

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

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

To find the opposite of xdy-dy^{2}, find the opposite of each term.

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

Combine ydx and -xdy to get 0.

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

Combine all terms containing d.

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

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

$$d=0$$

Solve for d

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

Steps for Solving Linear Equation

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

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

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

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

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

$$dx^{2}+ydx-\left(xd-yd\right)y=0$$

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

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

To find the opposite of xdy-dy^{2}, find the opposite of each term.

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

Combine ydx and -xdy to get 0.

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

Combine all terms containing d.

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

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

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