Solve dx/x^2-xy+y^2=dy/2y^2-xy

$$\frac { d x } { x ^ { 2 } - x y + y ^ { 2 } } = \frac { d y } { 2 y ^ { 2 } - x y }$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&y\neq 0\text{ and }x\neq 2y\\d\in \mathrm{R}\text{, }&\left(x=\frac{y}{2}\text{ or }x=y\right)\text{ and }y\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by y\left(x-2y\right)\left(-x^{2}+xy-y^{2}\right), the least common multiple of x^{2}-xy+y^{2},2y^{2}-xy.

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

Use the distributive property to multiply -y by x-2y.

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

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

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

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

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

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

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

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

$$-ydx^{2}+2y^{2}dx=x^{2}dy-xdy^{2}+dy^{3}$$

Subtract x^{2}dy from both sides.

$$-ydx^{2}+2y^{2}dx-x^{2}dy=-xdy^{2}+dy^{3}$$

Combine -ydx^{2} and -x^{2}dy to get -2ydx^{2}.

$$-2ydx^{2}+2y^{2}dx=-xdy^{2}+dy^{3}$$

Add xdy^{2} to both sides.

$$-2ydx^{2}+2y^{2}dx+xdy^{2}=dy^{3}$$

Combine 2y^{2}dx and xdy^{2} to get 3y^{2}dx.

$$-2ydx^{2}+3y^{2}dx=dy^{3}$$

Subtract dy^{3} from both sides.

$$-2ydx^{2}+3y^{2}dx-dy^{3}=0$$

Reorder the terms.

$$3dxy^{2}-dy^{3}-2dyx^{2}=0$$

Combine all terms containing d.

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

Divide 0 by 3xy^{2}-y^{3}-2yx^{2}.

$$d=0$$

Solve for x

$$\left\{\begin{matrix}x=y\text{; }x=\frac{y}{2}\text{, }&y\neq 0\\x\neq 2y\text{, }&d=0\text{ and }y\neq 0\end{matrix}\right.$$