# 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.