Existence and Uniqueness Theorem for ODEs
For y' = f(t, y), continuity of f guarantees a local solution and continuity of the partial derivative of f with respect to y (a Lipschitz condition) guarantees it is unique. Solution curves therefore cannot cross where the theorem's hypotheses hold, and failure of the hypotheses (e.g. y' = y^(1/3) at y = 0) permits multiple solutions.
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For y' = f(t, y), continuity of f guarantees a local solution and continuity of the partial derivative of f with respect to y (a Lipschitz condition) guarantees it is unique. Solution curves therefore cannot cross where the theorem's hypotheses hold, and failure of the hypotheses (e.g. y' = y^(1/3) at y = 0) permits multiple solutions.
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