y), and it can also be applied to systems of equations. Global uniqueness and maximum domain of solution When the hypotheses of the Picard–Lindelöf theorem are satisfied, then local existence and uniqueness can be extended to a global result. More precisely: For each initial condition (x0, y0) there exists a unique maximum (possibly infinite) open interval such that any solution that satisfies this initial condition is a restriction of the solution that satisfies this initial condition with domain . In the case that , there are exactly two possibilities explosion in finite time: leaves domain of definition: where Ω is the open set in which F is defined, and is its boundary. Note that the maximum domain of the solution is always an interval (to have uniqueness) may be smaller than may depend on the specific choice of (x0, y0). Example. This means that F(x, y) = y2, which is C1 and therefore locally Lipschitz continuous, satisfying the Picard–Lindelöf theorem. Even in such a simple setting, the maximum domain of solution cannot be all since the solution is which has maximum domain: This shows clearly that the maximum interval may depend on the initial conditions. The domain of y could be taken as being but this would lead to a domain that is not an interval, so that the side opposite to the initial condition would be disconnected from the initial condition, and therefore not uniquely determined by it. The maximum domain is not because which is one of the two possible cases according to the above theorem. Reduction of order Differential equations can usually be solved more easily if the order of the equation can be reduced. Reduction to a first-order system Any explicit differential equation of order n, can be written as a system of n first-order differential equations by defining a new family of unknown functions for i = 1, 2,..., n. The n-dimensional system of first-order coupled differential equations is then more compactly in vector notation: where Summary of exact solutions Some differential equations have solutions that can be written in an exact and closed form. Several important classes are given here. In the table below, , , , , and , are any integrable functions of , , and and are real given constants, and are arbitrary constants (complex in general). The differential equations are in their equivalent and alternative forms that lead to the