Help solving differential equation 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 :)
 A: Let $\omega = \sqrt{k/m}$ so that the differential equation we wish to solve is
$$ y'' + \omega^2 y = -G $$
Given any two solutions $y_1, y_2$, their difference satisfies
$y'' + \omega^2 y = 0$,
and the general solution of this is given by $y(t) = A cos(\omega t) + B \sin(\omega t)$. Hence the general solution of the original equation is
$$ y(t) = A \cos(\omega t) + B \sin (\omega t) + y_p(t)$$
where $y_p(t)$ is any particular solution to the original equation. However, the equation is easy enough that we can guess a solution using the method of undetermined coefficients. In this case we can guess $y_p(t) = c$ where $c$ is a constant. Plugging it into the differential equation yields:
$$ \omega^2 c = - G$$
hence $c = -G / \omega^2$. So the general solution to the original equation is
$$ y(t) = A \cos(\omega t) + B \sin(\omega t) - \frac{G}{\omega^2}$$
Setting $t=0$, we find
$$ y_0 = A - \frac{G}{\omega^2} $$
and
$$ y'_0 = B \omega $$
so that
$$ y(t) = \left(y_0 + \frac{G}{\omega^2} \right) \cos(\omega t) + \frac{y'_0}{\omega} \sin(\omega t) - \frac{G}{\omega^2} $$
A: I have not checked your derivation of the final differential equation, but assuming we start from
$$y''=-\frac k m y - G$$
Where $k,m,G$ are constants, we can use the annihilator method, considering the linear differential operator $L=D(D^2+\frac k m)$. This has the general solution $y(t)=-\frac k m G+c_1\cos{\sqrt \frac k m t}+c_2\sin{\sqrt\frac k m t}$, and the coefficients $c_1,c_2$ can easily be found by plugging in your initial conditions.
