Solve -k/x^2dx=mvdv

$$- \frac { k } { x ^ { 2 } } d x = m v d v$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=0\text{, }&x\neq 0\\d\in \mathrm{C}\text{, }&k=-mxv^{2}\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{2}.

$$\left(-\frac{k}{x^{2}}\right)dxx^{2}=mvdvx^{2}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\left(-\frac{k}{x^{2}}\right)dx^{3}=mvdvx^{2}$$

Multiply v and v to get v^{2}.

$$\left(-\frac{k}{x^{2}}\right)dx^{3}=mv^{2}dx^{2}$$

Express \left(-\frac{k}{x^{2}}\right)d as a single fraction.

$$\frac{-kd}{x^{2}}x^{3}=mv^{2}dx^{2}$$

Express \frac{-kd}{x^{2}}x^{3} as a single fraction.

$$\frac{-kdx^{3}}{x^{2}}=mv^{2}dx^{2}$$

Cancel out x^{2} in both numerator and denominator.

$$-dkx=mv^{2}dx^{2}$$

Subtract mv^{2}dx^{2} from both sides.

$$-dkx-mv^{2}dx^{2}=0$$

Reorder the terms.

$$-dmv^{2}x^{2}-dkx=0$$

Combine all terms containing d.

$$\left(-mv^{2}x^{2}-kx\right)d=0$$

Divide 0 by -mv^{2}x^{2}-kx.

$$d=0$$

Solve for k (complex solution)

$$\left\{\begin{matrix}k=-mxv^{2}\text{, }&x\neq 0\\k\in \mathrm{C}\text{, }&d=0\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{2}.

$$\left(-\frac{k}{x^{2}}\right)dxx^{2}=mvdvx^{2}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\left(-\frac{k}{x^{2}}\right)dx^{3}=mvdvx^{2}$$

Multiply v and v to get v^{2}.

$$\left(-\frac{k}{x^{2}}\right)dx^{3}=mv^{2}dx^{2}$$

Express \left(-\frac{k}{x^{2}}\right)d as a single fraction.

$$\frac{-kd}{x^{2}}x^{3}=mv^{2}dx^{2}$$

Express \frac{-kd}{x^{2}}x^{3} as a single fraction.

$$\frac{-kdx^{3}}{x^{2}}=mv^{2}dx^{2}$$

Cancel out x^{2} in both numerator and denominator.

$$-dkx=mv^{2}dx^{2}$$

The equation is in standard form.

$$\left(-dx\right)k=dmv^{2}x^{2}$$

Divide both sides by -dx.

$$\frac{\left(-dx\right)k}{-dx}=\frac{dmv^{2}x^{2}}{-dx}$$

Dividing by -dx undoes the multiplication by -dx.

$$k=\frac{dmv^{2}x^{2}}{-dx}$$

Divide mv^{2}dx^{2} by -dx.

$$k=-mxv^{2}$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&x\neq 0\\d\in \mathrm{R}\text{, }&k=-mxv^{2}\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{2}.

$$\left(-\frac{k}{x^{2}}\right)dxx^{2}=mvdvx^{2}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\left(-\frac{k}{x^{2}}\right)dx^{3}=mvdvx^{2}$$

Multiply v and v to get v^{2}.

$$\left(-\frac{k}{x^{2}}\right)dx^{3}=mv^{2}dx^{2}$$

Express \left(-\frac{k}{x^{2}}\right)d as a single fraction.

$$\frac{-kd}{x^{2}}x^{3}=mv^{2}dx^{2}$$

Express \frac{-kd}{x^{2}}x^{3} as a single fraction.

$$\frac{-kdx^{3}}{x^{2}}=mv^{2}dx^{2}$$

Cancel out x^{2} in both numerator and denominator.

$$-dkx=mv^{2}dx^{2}$$

Subtract mv^{2}dx^{2} from both sides.

$$-dkx-mv^{2}dx^{2}=0$$

Reorder the terms.

$$-dmv^{2}x^{2}-dkx=0$$

Combine all terms containing d.

$$\left(-mv^{2}x^{2}-kx\right)d=0$$

Divide 0 by -mv^{2}x^{2}-kx.

$$d=0$$

Solve for k

$$\left\{\begin{matrix}k=-mxv^{2}\text{, }&x\neq 0\\k\in \mathrm{R}\text{, }&d=0\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{2}.

$$\left(-\frac{k}{x^{2}}\right)dxx^{2}=mvdvx^{2}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\left(-\frac{k}{x^{2}}\right)dx^{3}=mvdvx^{2}$$

Multiply v and v to get v^{2}.

$$\left(-\frac{k}{x^{2}}\right)dx^{3}=mv^{2}dx^{2}$$

Express \left(-\frac{k}{x^{2}}\right)d as a single fraction.

$$\frac{-kd}{x^{2}}x^{3}=mv^{2}dx^{2}$$

Express \frac{-kd}{x^{2}}x^{3} as a single fraction.

$$\frac{-kdx^{3}}{x^{2}}=mv^{2}dx^{2}$$

Cancel out x^{2} in both numerator and denominator.

$$-dkx=mv^{2}dx^{2}$$

The equation is in standard form.

$$\left(-dx\right)k=dmv^{2}x^{2}$$

Divide both sides by -dx.

$$\frac{\left(-dx\right)k}{-dx}=\frac{dmv^{2}x^{2}}{-dx}$$

Dividing by -dx undoes the multiplication by -dx.

$$k=\frac{dmv^{2}x^{2}}{-dx}$$

Divide mv^{2}dx^{2} by -dx.

$$k=-mxv^{2}$$