Solve (mg-kv)dt=dv

$$( m g - k v ) d t = d v$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(t=0\text{ and }v=0\right)\text{ or }\left(v=0\text{ and }g=0\text{ and }t\neq 0\right)\text{ or }\left(k=-\frac{1}{t}\text{ and }t\neq 0\text{ and }g=0\right)\text{ or }\left(m=\frac{v\left(kt+1\right)}{gt}\text{ and }t\neq 0\text{ and }g\neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply mg-kv by d.

$$\left(mgd-kvd\right)t=dv$$

Use the distributive property to multiply mgd-kvd by t.

$$mgdt-kvdt=dv$$

Subtract dv from both sides.

$$mgdt-kvdt-dv=0$$

Combine all terms containing d.

$$\left(mgt-kvt-v\right)d=0$$

The equation is in standard form.

$$\left(gmt-ktv-v\right)d=0$$

Divide 0 by mgt-kvt-v.

$$d=0$$

Solve for g

$$\left\{\begin{matrix}g=\frac{v\left(kt+1\right)}{mt}\text{, }&t\neq 0\text{ and }m\neq 0\\g\in \mathrm{R}\text{, }&\left(v=0\text{ and }t=0\right)\text{ or }\left(v=0\text{ and }m=0\text{ and }t\neq 0\right)\text{ or }\left(k=-\frac{1}{t}\text{ and }t\neq 0\text{ and }m=0\right)\text{ or }d=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply mg-kv by d.

$$\left(mgd-kvd\right)t=dv$$

Use the distributive property to multiply mgd-kvd by t.

$$mgdt-kvdt=dv$$

Add kvdt to both sides.

$$mgdt=dv+kvdt$$

The equation is in standard form.

$$dmtg=dktv+dv$$

Divide both sides by mdt.

$$\frac{dmtg}{dmt}=\frac{dv\left(kt+1\right)}{dmt}$$

Dividing by mdt undoes the multiplication by mdt.

$$g=\frac{dv\left(kt+1\right)}{dmt}$$

Divide vd\left(1+kt\right) by mdt.

$$g=\frac{v\left(kt+1\right)}{mt}$$