Solve m*g-k*(df(t)/dx)^2=m*d^2f(x)/dx^2

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

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}(ft)}{\mathrm{d}x}\right)^{2} to both sides.

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

The equation is in standard form.

$$mg=0$$

Divide 0 by m.

$$g=0$$

Solve for f

$$f\in \mathrm{R}$$

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