Uniqueness of Continuously Differentiable Solution to Initial Value Problem

Theorem

Let $D \subseteq \R^2$ be a region containing $\tuple {a, b}$.

Let $f: D \to \R$ be real-valued mapping such that $f$ and $\dfrac {\partial f} {\partial x}$ are continuous on $D$.

Consider the initial value problem:

$\dfrac {\d x} {\d t} = \map f {t, x}$

$\map x a = b$

Suppose the initial value problem above has a solution $\phi$ for all $x$ in some interval $J$ containing $a$.

Then the solution $\phi$ is unique on $J$.

Proof

This theorem requires a proof. In particular: Is this the same as Picard's Little Theorem? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 2007: Richard K. Miller and Anthony N. Michel: Ordinary Differential Equations: $\S 2.4.3$