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where ''f'' is a function that maps [''t''<sub>0</sub>,∞) × '''R'''<sup>d</sup> to '''R'''<sup>d</sup>, and the initial condition ''y''<sub>0</sub> ∈ '''R'''<sup>d</sup> is a given vector.
The above formulation is called an ''[[initial value problem]]'' (IVP). The [[Picard-Lindelöf theorem]] states that there is a unique solution, if ''f'' is [[Lipschitz continuous]]. In contrast, ''[[boundary value problem]]s'' (BVPs) specify (components of) the solution ''y'' at more than one
Note that we restrict ourselves to ''first-order'' differential equations (meaning that only the first derivative of ''y'' appears in the equation, and no higher derivatives). However, a higher-order equation can easily be converted to a first-order equation by introducing extra variables. For example, the second-order equation ''y''<nowiki>''</nowiki> = −''y'' can be rewritten as two first-order equations: ''y''<nowiki>'</nowiki> = ''z'' and ''z''<nowiki>'</nowiki> = −''y''.
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