Question: Consider an ODE \(\frac{d y}{d x}=f(x, y)\) where \(f\) is a function satisfying the Lipschitz conditions on a domain \(D \subset \mathbb{R}^{2}\). Show that if \(\phi_{1}(x)\) and \(\phi_{2}(x)\) are two distinct continuous solutions of the ODE on \(D\), then they cannot cross, i.e. there is no \(x_{0} \in D\) such that \(\phi_{1}\left(x_{0}\right) \quad \phi_{2}\left(x_{0}\right) .\)

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