A **system of differential equations** is a finite set of [[Differential Equation|differential equations]]. # Properties When do systems have solutions? Clearly overdetermined if there are more equations than dependent variables. What is meant by a general solution. Solutions will always have a general form. System itself is an implicit general form. Determined = Unique direction field? # Applications **High order scalar ODEs**: Note that [[Reduction of Order of Scalar Ordinary Differential Equations|reduction of order]] asserts that any [[Ordinary Differential Equation|scalar ordinary differential equation]] with order $n\geq 2$ can be written as an [[System of First Order Ordinary Differential Equations|n x n system of first order equations]]. In particular, a homogenous second order linear scalar ODE with real coefficients can be written as a [[Solution to Homogenous Linear 2 x 2 System of First Order Ordinary Differential Equations with Real Coefficients|2 x 2 system of homogenous linear first order ordinary differential equations]] with constant coefficients.