Solve d^2v/dt^2=2wg-4w^2v

$$\frac { d ^ { 2 } v } { d t ^ { 2 } } = 2 w g - 4 w ^ { 2 } v$$

Solve for g

$$\left\{\begin{matrix}\\g=2vw\text{, }&\text{unconditionally}\\g\in \mathrm{R}\text{, }&w=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$2wg-4w^{2}v=\frac{\mathrm{d}(v)}{\mathrm{d}t^{2}}$$

Add 4w^{2}v to both sides.

$$2wg=\frac{\mathrm{d}(v)}{\mathrm{d}t^{2}}+4w^{2}v$$

The equation is in standard form.

$$2wg=4vw^{2}$$

Divide both sides by 2w.

$$\frac{2wg}{2w}=\frac{4vw^{2}}{2w}$$

Dividing by 2w undoes the multiplication by 2w.

$$g=\frac{4vw^{2}}{2w}$$

Divide 4w^{2}v by 2w.

$$g=2vw$$

Solve for t

$$t\in \mathrm{R}$$

$$w=0\text{ or }v=\frac{g}{2w}$$