Flow of a differential equation over what interval

Question

Let $\dot{x}=x^2$. Over what interval is the flow defined?

I can see that the solution is of the initial value problem $\dot{x}=x^2$, $x(0)=x_0\ $ is $$ x(t)=\frac{x_0}{1-x_0\cdot t}$$ and that it blows up at $T=\frac{1}{x_0}$.

However I cannot see how you say what interval to flow is defined over and for what reasons.

Answer 1

The flow is

$$\phi(t,x) = \frac{x}{1 - t \cdot x}$$

defined on the subset of $\mathbb{R}^2$ $\{(t,x)\ \mid \ 1 - t\cdot x \ne 0\}$, the complement of the hyperbola $t x = 1$. For every $x$ in $\mathbb{R}$ horizontal line through $(0,x)$ may intersect the hyperbola in $1$ or zero (for $x=0$ ) points. Consider the on that line cut by the hyperbola and containing $(x,0)$, its projection on the $t$ axis is the maximum domain of definition of your solution.

There are $3$ cases, $x>0$, $x=0$, $x<0$.

1. For $x> 0$ the domain is $(-\infty, \frac{1}{x})$

2. For $x=0$ the domain is $\mathbb{R}$.

3. For $x < 0$ the domain is $( \frac{1}{x}, \infty)$

Note: In all of these cases the domain is an interval containing $0$.