Case I: The force acting on an object of mass m is $F(x) = F\_o(1-e^{\alpha x})$

Case II: The force acting on an object of mass m is $F(x) = F\_o(1-e^{-\alpha x})$

where $F\_o$ and $\alpha$ are positive constants. In which case can the object oscillate 
near the equilibrium and what is the oscillation frequency?

SOLVING FOR CASE I

The total force on the object is: $$ F(x, x', t) = -kx + F\_o(1-e^{\alpha x})$$
and can be rewritten as: $$ x'' + w^2 x = F\_o(1-e^{\alpha x})$$
where $w^2 = -k/m$

This is an inhomogeneous linear differential equation. I first solve for the complementary solution, $x\_c$, for $x'' + w^2x = 0$. This gives me: $$x\_c = Ae^{0x} + Be^{-w^2x} = A + Be^{-w^2x}$$

I can then determine the particular solution, $x\_p$ for: 
$$ 1)x'' + w^2 x = F\_o$$ 
$$ x\_{p1} = \frac{F\_o}{w^2} $$
$$2) x'' + w^2 x = -F\_oe^{\alpha x}$$
$$ x\_{p2} = \frac{-F\_oe^{\alpha x}}{\alpha ^2 + 1}$$

For a general solution of:
$$ x = A + Be^{-w^2 x} + \frac{F\_o}{w^2} + \frac{-F\_oe^{\alpha x}}{\alpha ^2 + 1}$$

This raises two questions:

1 - Is this solution correct?

2- Once I solve for $F(x) = F\_o(1-e^{-\alpha x})$, how do I determine the case in which the object oscillates near the equilibrium? I suppose I would need to minimize the amplitude? I'm not sure how to approach this.