Let $$\ddot{x}+c \dot{x}+x(1-x)=0 $$ Find sufficient conditions for $c$ such that the solution follows this properties:
$$\lim_{t \to -\infty}x(t)=0 \ \ \ \lim_{t \to \infty}x(t)=1 \ \ \ \dot{x}>0 \ \ \forall t$$
My attempt: I change the differential equation into a system of differential equations. I have the following:
\begin{align} \dot{x}_1 &= x_2\\ \dot{x}_2 &= -cx_2-x_1+x_1^2. \end{align}
Then, I calculate the equilibrium and get $P_1 = (0,0)$ and $P_2 = (1,0)$. The problem arises when I analyze the stability. For $P_1$ the point must be inestable and for $P_2$ stable. But I have some problems with the characteristic polynomial. For $P_1$ I got $$\lambda^2+c\lambda+1,$$ and for $P_2$ I got $$\lambda^2+c\lambda-1.$$ However, I do not know how I could assure both conditions in my problem.
Any help?