- #1

MathStudent999

- 1

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- TL;DR Summary
- I was hoping to find more clarification on uniqueness results for autonomous ODEs

I'm studying ODEs and have understood most of the results of the first chapter of my ODE book, this is still bothers me. Suppose

$$\begin{cases}

f \in \mathcal{C}(\mathbb{R}) \\

\dot{x} = f(x) \\

x(0) = 0 \\

f(0) = 0 \\

\end{cases}.

$$

Then,

$$

\lim_{\varepsilon \searrow 0}\left|\int^\varepsilon_0 {dy \over f(y)}\right| = \infty

$$

implies solutions are unique. Since

$$

\lim_{\varepsilon \searrow 0}\left|\int^\varepsilon_0 {dy \over f(y)}\right|< \infty

$$

allows us to invert to get a solution(more clarification on this) other than 0. So, am I seeing it right that this is just a contrapostive to get uniqueness.

$$\begin{cases}

f \in \mathcal{C}(\mathbb{R}) \\

\dot{x} = f(x) \\

x(0) = 0 \\

f(0) = 0 \\

\end{cases}.

$$

Then,

$$

\lim_{\varepsilon \searrow 0}\left|\int^\varepsilon_0 {dy \over f(y)}\right| = \infty

$$

implies solutions are unique. Since

$$

\lim_{\varepsilon \searrow 0}\left|\int^\varepsilon_0 {dy \over f(y)}\right|< \infty

$$

allows us to invert to get a solution(more clarification on this) other than 0. So, am I seeing it right that this is just a contrapostive to get uniqueness.