existence and uniqueness theorem

Also known as the Picard-Lindelof theorem or the Cauchy-Lipschitz theorem

A differential equation of the form

\begin{equation} \dot{y} = f(x, y) y(x_0) = y_0 \end{equation}

Has a unique solution \(y\) on the interval \((x_0 - a, x_0 + a)\) with \(a > 0\) if \(f: \mathcal{D} \to \mathbb{R}\) where \(\mathcal{D}\) is an open set that contains \((x_0, y_0)\) is lipschitz continuous.

The larger the lipschitz constant, the smaller this bound for a unique solution becomes.

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