Solve mg-k(dx/dt)^2=md^2x/dt^2

$$m g - k ( \frac { d x } { d t } ) ^ { 2 } = m \frac { d ^ { 2 } x } { d t ^ { 2 } }$$

Solve for g

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

Steps for Solving Linear Equation

Add k\left(\frac{\mathrm{d}(x)}{\mathrm{d}t}\right)^{2} to both sides.

$$mg=m\frac{\mathrm{d}(x)}{\mathrm{d}t^{2}}+k\left(\frac{\mathrm{d}(x)}{\mathrm{d}t}\right)^{2}$$

The equation is in standard form.

$$mg=0$$

Divide 0 by m.

$$g=0$$

Solve for k

$$k\in \mathrm{R}$$

$$g=0\text{ or }m=0$$

Steps for Solving Linear Equation

Subtract mg from both sides.

$$-k\left(\frac{\mathrm{d}(x)}{\mathrm{d}t}\right)^{2}=m\frac{\mathrm{d}(x)}{\mathrm{d}t^{2}}-mg$$

The equation is in standard form.

$$0=-gm$$

This is false for any k.

$$k\in$$