Discovering the General Solution of ODEs: A Comprehensive Guide

[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:

[tex]x' = x(1-x)[/tex]

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

note:
The definition of 'general solution' I'm referring to here is the following:

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

[tex]x' = f(t,x)[/tex]

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

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

Any recommended examples/literature?

 

[gtoguy97]

In the case of the differential equation x' = x(1-x), the general solution is given by \lambda_{max}(t, \tau,\xi) = \frac{\xi e^{(t-\tau)}}{1 + \xi (e^{(t-\tau)} - 1)}, and the domain of definition is \Omega := \left\{(t, \tau,\xi)\inR^{1+1+1} : (\tau,\xi)\in [0,\infty)\times [0,1], t \in [\tau, +\infty)\right\}. For more information on general solutions and related topics, you may find the following references useful:1. A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, 2003.2. E. I. Khukhro, Ordinary Differential Equations: Introduction with Applications, Cambridge University Press, 2014.3. S. H. S. Kazmi, Differential Equations: Theory, Technique, and Practice, Elsevier, 2016.