Differential Equation for a schematic harmonic oscillator.

Question

Differential Equation for a schematic harmonic oscillator.

here are some pictures that i tried to solve but i don't know it is true or completely wrong.

Answer 1

Hint.

Calling $X = x - x_0$ with $x_0 = \frac{m g}{k}$ we have

$$ m\ddot X = F - c\dot X - k X $$