Ok people, trying to fix my question before I have the following problem:
I need to find $ x(t) $ so I can evaluate it in $t=t_f$ integrating (with respect to $t$) $\int_0^{t_f}\int \ddot x(t) \, dt$ either numerically or not.
$\ddot{x} = x\dot{\alpha}^2 + k$ ; $k={}$constant
and
$\dot{\alpha}(t) = a+ bt $ ; $a,b ={}$constants
so in other words I need to integrate
$\ddot{x} = x( a+ bt )^2 + k$
do I need extra information to solve this or it can be done? thanks a lot.