# General solution of ODE

1. Jun 23, 2009

### h0h0

The flow defined by a differential equation/general solution

I'm trying to find the 'general solution'(perhaps more accurately the flow)
of the following ODE:

$$x' = x(1-x)$$

i.e. $$\lambda_{max}(t, \tau,\xi)$$ and the domain where it's defined $$\Omega$$

note:
The definition of 'general solution' i'm refering to here is the following:

Given a open $$D\subset R^{1+N}$$, a continous and with respect to x Lipschitz-continous function $$f:D \rightarrow R^{N}$$, and the differential equation

$$x' = f(t,x)$$

The function $$\lambda(t, \tau,\xi) := \lambda_{max}(t, \tau,\xi)$$ defined for $$(t, \tau,\xi)\in\Omega := \left\{(t, \tau,\xi)\inR^{1+1+N} : (\tau,\xi)\in D, t \in I_{max}(\tau,\xi)\right\}$$
is called the 'general solution' of the differential equation.

$$\lambda_{max}$$ is the maximal solution, and $$I_{max}(\tau,\xi)$$ the maximal interval of
existence of the solution of the intial value problem $$x(\tau)=\xi$$

Any recommended examples/literature?

Last edited: Jun 23, 2009