Cauchy-Lipschitz theorem

The Cauchy-Lipschitz theorem, also known as the Picard-Lindelöf theorem, ensures that for certain types of differential equations, there is a unique solution passing through a given point. Specifically, if the function defining the equation is continuous and doesn’t change too rapidly (Lipschitz continuous), then starting from an initial value, the solution behaves predictably and is guaranteed to exist and be unique within a small interval. This theorem provides a foundation for understanding how differential equations model real-world systems, confirming that their solutions are well-defined and dependable under the right conditions.