Solve vdv=adx

$$v d v = a d x$$

Solve for a (complex solution)

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

Steps for Solving Linear Equation

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

$$v^{2}d=adx$$

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

$$adx=v^{2}d$$

The equation is in standard form.

$$dxa=dv^{2}$$

Divide both sides by dx.

$$\frac{dxa}{dx}=\frac{dv^{2}}{dx}$$

Dividing by dx undoes the multiplication by dx.

$$a=\frac{dv^{2}}{dx}$$

Divide v^{2}d by dx.

$$a=\frac{v^{2}}{x}$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

$$v^{2}d=adx$$

Subtract adx from both sides.

$$v^{2}d-adx=0$$

Reorder the terms.

$$-adx+dv^{2}=0$$

Combine all terms containing d.

$$\left(-ax+v^{2}\right)d=0$$

The equation is in standard form.

$$\left(v^{2}-ax\right)d=0$$

Divide 0 by v^{2}-ax.

$$d=0$$

Solve for a

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

Steps for Solving Linear Equation

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

$$v^{2}d=adx$$

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

$$adx=v^{2}d$$

The equation is in standard form.

$$dxa=dv^{2}$$

Divide both sides by dx.

$$\frac{dxa}{dx}=\frac{dv^{2}}{dx}$$

Dividing by dx undoes the multiplication by dx.

$$a=\frac{dv^{2}}{dx}$$

Divide v^{2}d by dx.

$$a=\frac{v^{2}}{x}$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(v=0\text{ and }a=0\text{ and }x=0\right)\text{ or }\left(a\geq 0\text{ and }x\geq 0\text{ and }|v|=\sqrt{ax}\right)\text{ or }\left(x\leq 0\text{ and }a\leq 0\text{ and }|v|=\sqrt{ax}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$v^{2}d=adx$$

Subtract adx from both sides.

$$v^{2}d-adx=0$$

Reorder the terms.

$$-adx+dv^{2}=0$$

Combine all terms containing d.

$$\left(-ax+v^{2}\right)d=0$$

The equation is in standard form.

$$\left(v^{2}-ax\right)d=0$$

Divide 0 by v^{2}-ax.

$$d=0$$