Solve mdv/dt=-ky+mg

$$m \frac { d v } { d t } = - k y + m g$$

Solve for g

$$\left\{\begin{matrix}g=\frac{ky}{m}\text{, }&m\neq 0\\g\in \mathrm{R}\text{, }&\left(k=0\text{ or }y=0\right)\text{ and }m=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(-k\right)y+mg=m\frac{\mathrm{d}}{\mathrm{d}t}(v)$$

Subtract \left(-k\right)y from both sides.

$$mg=m\frac{\mathrm{d}}{\mathrm{d}t}(v)-\left(-k\right)y$$

Multiply -1 and -1 to get 1.

$$mg=m\frac{\mathrm{d}}{\mathrm{d}t}(v)+ky$$

The equation is in standard form.

$$mg=ky$$

Divide both sides by m.

$$\frac{mg}{m}=\frac{ky}{m}$$

Dividing by m undoes the multiplication by m.

$$g=\frac{ky}{m}$$

Solve for k

$$\left\{\begin{matrix}k=\frac{gm}{y}\text{, }&y\neq 0\\k\in \mathrm{R}\text{, }&\left(m=0\text{ or }g=0\right)\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(-k\right)y+mg=m\frac{\mathrm{d}}{\mathrm{d}t}(v)$$

Subtract mg from both sides.

$$\left(-k\right)y=m\frac{\mathrm{d}}{\mathrm{d}t}(v)-mg$$

Reorder the terms.

$$-ky=m\frac{\mathrm{d}}{\mathrm{d}t}(v)-gm$$

The equation is in standard form.

$$\left(-y\right)k=-gm$$

Divide both sides by -y.

$$\frac{\left(-y\right)k}{-y}=-\frac{gm}{-y}$$

Dividing by -y undoes the multiplication by -y.

$$k=-\frac{gm}{-y}$$

Divide -gm by -y.

$$k=\frac{gm}{y}$$