Let be continuous functions on an open interval J and let L be the linear differential operator
where and . Let and let be n given real numbers. Then there exists 1 and only 1 function that satisfies both the homogeneous equation on J and the initial conditions
Proof of this will be obtained as a
The initial conditions can be represented by a vector in Rn with components
This is also called the initial- value vector of f at x0. The uniqueness theorem then implies only can satisfy an initial-value vector that equals to O.