Solve ydx-4xdy=y^6dy

$$y d x - 4 x d y = y ^ { 6 } d y$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=-\frac{y^{6}}{3}\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

To multiply powers of the same base, add their exponents. Add 6 and 1 to get 7.

$$ydx-4xdy=y^{7}d$$

Combine ydx and -4xdy to get -3ydx.

$$-3ydx=y^{7}d$$

Subtract y^{7}d from both sides.

$$-3ydx-y^{7}d=0$$

Reorder the terms.

$$-3dxy-dy^{7}=0$$

Combine all terms containing d.

$$\left(-3xy-y^{7}\right)d=0$$

Divide 0 by -3xy-y^{7}.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-\frac{y^{6}}{3}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

To multiply powers of the same base, add their exponents. Add 6 and 1 to get 7.

$$ydx-4xdy=y^{7}d$$

Combine ydx and -4xdy to get -3ydx.

$$-3ydx=y^{7}d$$

The equation is in standard form.

$$\left(-3dy\right)x=dy^{7}$$

Divide both sides by -3yd.

$$\frac{\left(-3dy\right)x}{-3dy}=\frac{dy^{7}}{-3dy}$$

Dividing by -3yd undoes the multiplication by -3yd.

$$x=\frac{dy^{7}}{-3dy}$$

Divide y^{7}d by -3yd.

$$x=-\frac{y^{6}}{3}$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=-\frac{y^{6}}{3}\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

To multiply powers of the same base, add their exponents. Add 6 and 1 to get 7.

$$ydx-4xdy=y^{7}d$$

Combine ydx and -4xdy to get -3ydx.

$$-3ydx=y^{7}d$$

Subtract y^{7}d from both sides.

$$-3ydx-y^{7}d=0$$

Reorder the terms.

$$-3dxy-dy^{7}=0$$

Combine all terms containing d.

$$\left(-3xy-y^{7}\right)d=0$$

Divide 0 by -3xy-y^{7}.

$$d=0$$

Solve for x

$$\left\{\begin{matrix}\\x=-\frac{y^{6}}{3}\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

To multiply powers of the same base, add their exponents. Add 6 and 1 to get 7.

$$ydx-4xdy=y^{7}d$$

Combine ydx and -4xdy to get -3ydx.

$$-3ydx=y^{7}d$$

The equation is in standard form.

$$\left(-3dy\right)x=dy^{7}$$

Divide both sides by -3yd.

$$\frac{\left(-3dy\right)x}{-3dy}=\frac{dy^{7}}{-3dy}$$

Dividing by -3yd undoes the multiplication by -3yd.

$$x=\frac{dy^{7}}{-3dy}$$

Divide y^{7}d by -3yd.

$$x=-\frac{y^{6}}{3}$$