Solve dv(t)/dt=-bv(t)/m-g

$$\frac { d v ( t ) } { d t } = - \frac { b v ( t ) } { m } - g$$

Solve for b

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

Steps for Solving Linear Equation

Multiply both sides of the equation by m.

$$m\frac{\mathrm{d}}{\mathrm{d}t}(vt)=-bvt-gm$$

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

$$-bvt-gm=m\frac{\mathrm{d}}{\mathrm{d}t}(vt)$$

Add gm to both sides.

$$-bvt=m\frac{\mathrm{d}}{\mathrm{d}t}(vt)+gm$$

Reorder the terms.

$$-btv=m\frac{\mathrm{d}}{\mathrm{d}t}(tv)+gm$$

The equation is in standard form.

$$\left(-tv\right)b=gm+mv$$

Divide both sides by -tv.

$$\frac{\left(-tv\right)b}{-tv}=\frac{m\left(v+g\right)}{-tv}$$

Dividing by -tv undoes the multiplication by -tv.

$$b=\frac{m\left(v+g\right)}{-tv}$$

Divide m\left(v+g\right) by -tv.

$$b=-\frac{m\left(v+g\right)}{tv}$$

Solve for g

$$g=-\frac{v\left(m+bt\right)}{m}$$

$$m\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by m.

$$m\frac{\mathrm{d}}{\mathrm{d}t}(vt)=-bvt-gm$$

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

$$-bvt-gm=m\frac{\mathrm{d}}{\mathrm{d}t}(vt)$$

Add bvt to both sides.

$$-gm=m\frac{\mathrm{d}}{\mathrm{d}t}(vt)+bvt$$

The equation is in standard form.

$$\left(-m\right)g=mv+btv$$

Divide both sides by -m.

$$\frac{\left(-m\right)g}{-m}=\frac{v\left(m+bt\right)}{-m}$$

Dividing by -m undoes the multiplication by -m.

$$g=\frac{v\left(m+bt\right)}{-m}$$

Divide v\left(m+bt\right) by -m.

$$g=-\frac{v\left(m+bt\right)}{m}$$