# Uniqueness of Continuously Differentiable Solution to Initial Value Problem

Jump to navigation
Jump to search

## 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.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*: $\S 2.4.3$