Maximal Existence Interval for DE with Continuous Derivative

[onie mti]

I am given that a DE with the form x' = f(x) is defined on the interval (c,b) where f has continuous derivative on its domain
How do i show that if f(p) = f(q) = 0 and x(t) is between p and q then the maximal interval of existence of x is (-∞, ∞)

 

[chisigma]

I am given that a DE with the form x' = f(x) is defined on the interval (c,b) where f has continuous derivative on its domain
How do i show that if f(p) = f(q) = 0 and x(t) is between p and q then the maximal interval of existence of x is (-∞, ∞)

Lets suppose that in p < x < q is f(x)>0 and no other zero of f(x) exists outside this interval. In thi case x=p is a repulsive fixed point, i.e. all the solution of the DE with the only exception of the constant solution x=p will diverge from p, and x=q is an attractive fixed point, i.e. all the solution of the DE with the only exception of the constant solution x=p will converge to q. That means that any solution of the DE with initial condition $x(t_{0})= x_{0},\ p < x_{0} < q$ will converge to q. Furthemore f(x) is function of the x alone, so that a if x(t) is solution of the DE, then $x(t + \tau),\ \tau \in \mathbb{R}$ is also solution of the DE. All that is sufficient to prove the hypothesis...

Kind regards

$\chi$ $\sigma$