Solve (x-2y)dy=2ydx

$$( x - 2 y ) d y = 2 y d x$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

$$\left(xd-2yd\right)y=2ydx$$

Use the distributive property to multiply xd-2yd by y.

$$xdy-2dy^{2}=2ydx$$

Subtract 2ydx from both sides.

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

Combine xdy and -2ydx to get -xdy.

$$-xdy-2dy^{2}=0$$

Combine all terms containing d.

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

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

$$d=0$$

Solve for x (complex solution)

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

Steps for Solving Linear Equation

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

$$\left(xd-2yd\right)y=2ydx$$

Use the distributive property to multiply xd-2yd by y.

$$xdy-2dy^{2}=2ydx$$

Subtract 2ydx from both sides.

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

Combine xdy and -2ydx to get -xdy.

$$-xdy-2dy^{2}=0$$

Add 2dy^{2} to both sides. Anything plus zero gives itself.

$$-xdy=2dy^{2}$$

The equation is in standard form.

$$\left(-dy\right)x=2dy^{2}$$

Divide both sides by -dy.

$$\frac{\left(-dy\right)x}{-dy}=\frac{2dy^{2}}{-dy}$$

Dividing by -dy undoes the multiplication by -dy.

$$x=\frac{2dy^{2}}{-dy}$$

Divide 2dy^{2} by -dy.

$$x=-2y$$

Solve for d

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

Steps for Solving Linear Equation

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

$$\left(xd-2yd\right)y=2ydx$$

Use the distributive property to multiply xd-2yd by y.

$$xdy-2dy^{2}=2ydx$$

Subtract 2ydx from both sides.

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

Combine xdy and -2ydx to get -xdy.

$$-xdy-2dy^{2}=0$$

Combine all terms containing d.

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

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

$$d=0$$

Solve for x

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

Steps for Solving Linear Equation

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

$$\left(xd-2yd\right)y=2ydx$$

Use the distributive property to multiply xd-2yd by y.

$$xdy-2dy^{2}=2ydx$$

Subtract 2ydx from both sides.

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

Combine xdy and -2ydx to get -xdy.

$$-xdy-2dy^{2}=0$$

Add 2dy^{2} to both sides. Anything plus zero gives itself.

$$-xdy=2dy^{2}$$

The equation is in standard form.

$$\left(-dy\right)x=2dy^{2}$$

Divide both sides by -dy.

$$\frac{\left(-dy\right)x}{-dy}=\frac{2dy^{2}}{-dy}$$

Dividing by -dy undoes the multiplication by -dy.

$$x=\frac{2dy^{2}}{-dy}$$

Divide 2dy^{2} by -dy.

$$x=-2y$$