The Existence and Uniqueness Theorem
for an
nth Order Linear Ordinary Differential Equation
We assume that n is a positive integer, α , β , t 0 , p0 , p1, p 2 , … , p n −1 are given real numbers with
α < t 0 < β , and we let I be the interval: I = ( α , β ).
Consider the nth order linear ordinary differential equation together with n initial conditions as follows:

dn y
d n −1 y
dy
+ a n (t) y = g (t)
 a 0 (t) n + a1 (t) n −1 + + a n −1 (t)
dt
dt
dt




y (t 0 ) = p0 , y ′ (t 0 ) = p1, , y(n −1) (t 0 ) = p n −1

We assume that a 0 (t), a1 (t),
a 0 (t) ≠ 0 for all t in I.







(1)
, a n −1 (t), a n (t) and g (t) are continuous functions of t on I, and
Then there exists a unique solution, y = φ (t) , of the Initial Value Problem (1) on the interval I.
This solution, y = φ (t) , will be continuous, will have continuous derivatives up to and including the nth order,
and will satisfy the differential equation at all points of the interval I. Thus, the above theorem is a Global
Existence and Uniqueness Theorem.