Solve (x-2y)y^prime=2x-yE

$$( x - 2 y ) y ^ { \prime } = 2 x - y E$$

Solve for E (complex solution)

$$\left\{\begin{matrix}E=\frac{2x}{y}\text{, }&y\neq 0\\E\in \mathrm{C}\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x-2y by \frac{\mathrm{d}}{\mathrm{d}x}(y).

$$x\frac{\mathrm{d}}{\mathrm{d}x}(y)-2y\frac{\mathrm{d}}{\mathrm{d}x}(y)=2x-yE$$

Swap sides so that all variable terms are on the left hand side.

$$2x-yE=x\frac{\mathrm{d}}{\mathrm{d}x}(y)-2y\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Subtract 2x from both sides.

$$-yE=x\frac{\mathrm{d}}{\mathrm{d}x}(y)-2y\frac{\mathrm{d}}{\mathrm{d}x}(y)-2x$$

The equation is in standard form.

$$\left(-y\right)E=-2x$$

Divide both sides by -y.

$$\frac{\left(-y\right)E}{-y}=-\frac{2x}{-y}$$

Dividing by -y undoes the multiplication by -y.

$$E=-\frac{2x}{-y}$$

Divide -2x by -y.

$$E=\frac{2x}{y}$$

Solve for E

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

Steps for Solving Linear Equation

Use the distributive property to multiply x-2y by \frac{\mathrm{d}}{\mathrm{d}x}(y).

$$x\frac{\mathrm{d}}{\mathrm{d}x}(y)-2y\frac{\mathrm{d}}{\mathrm{d}x}(y)=2x-yE$$

Swap sides so that all variable terms are on the left hand side.

$$2x-yE=x\frac{\mathrm{d}}{\mathrm{d}x}(y)-2y\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Subtract 2x from both sides.

$$-yE=x\frac{\mathrm{d}}{\mathrm{d}x}(y)-2y\frac{\mathrm{d}}{\mathrm{d}x}(y)-2x$$

The equation is in standard form.

$$\left(-y\right)E=-2x$$

Divide both sides by -y.

$$\frac{\left(-y\right)E}{-y}=-\frac{2x}{-y}$$

Dividing by -y undoes the multiplication by -y.

$$E=-\frac{2x}{-y}$$

Divide -2x by -y.

$$E=\frac{2x}{y}$$

Solve for x

$$x=\frac{Ey}{2}$$