I want to solve the following differential equation:
$y[t]$ : vertical position (height) of the object at time t
$y_c$ : height of the ceiling
$y_e$ : equilibrium point, the height at which the mass will stop at the end of its movement.
$a[t]$ : acceleration at time t
$t$ : time
$k$ : spring coefficient
$m$ : mass of the object
$G$ : gravity
$$\begin{align}
&F = -ky[t] - mG \\
\Leftrightarrow &ma[t] = -ky[t] -mG \\
\Leftrightarrow&my''[t] = -ky[t] -mG\\
\Leftrightarrow&y''[t] = -\dfrac{k}{m}y[t] -G
\end{align}$$
subject to:
$$y[0] = y_c\\y'[0] = 0$$
I'm not really sure how to do so. The equation is for modeling the movement of a falling object that is attached to a spring that is attached to the ceiling, thus gravity ($G$) is involved. appreciate your help:
Appreciate your help :)