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In some situations, the function ƒ is not of [[Smooth function|class ''C''<sup>1</sup>]], or even [[Lipschitz continuity|Lipschitz]], so the usual result guaranteeing the local existence of a unique solution does not apply. The [[Peano existence theorem]] however proves that even for ƒ merely continuous, solutions are guaranteed to exist locally in time; the problem is that there is no guarantee of uniqueness. The result may be found in Coddington & Levinson (1955, Theorem 1.3) or Robinson (2001, Theorem 2.6). An even more general result is the [[Carathéodory existence theorem]], which proves existence for some discontinuous functions ƒ.
Ways
A simple example is to solve <math>y' = 0.85 y</math> and <math>y(0) = 19</math>. We are trying to find a formula for <math>y(t)</math> that satisfies these two equations.
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