1550078481-Ordinary_Differential_Equations__Roberts_

N-th Order Linear Differential Equations 165

DEFINITION Initial Value Problem

An initial value problem for an n-th order linear differential equation

consists of solving the differential equation

subject to a set of n constraints, called initial conditions, of the form

(2) y ( ) xo _ - c1, y (1)( xo ) = c2, ... , y (n-1)( xo ) = Cn

where c1, c2, ... , Cn are any specified constants and xo is some specified point.

In section 2.2, we stated and proved an existence and uniqueness theorem
for the first-order linear initial value problem: y' = f(x, y); y(c) = d. The
following theorem, which we state without proof, generalizes this theorem to
n-th order linear initial value problems.

AN EXISTENCE AND UNIQUENESS THEOREM

If the functions an(x), an_ 1 (x), ... , a 1 (x), a 0 (x) and b(x) are all contin-

uous on an interval I and if an ( x) -j. 0 for any x in I, then there exists a

unique solution on I to the initial value problem consisting of the linear n-th

order differential equation

and the initial conditions

(2)

where c 1 , c 2 , ... , Cn are any specified constants and xo is some point in the

interval J.

Observe that the existence and uniqueness theorem guarantees that the
zero function, y(x) = 0, is the unique solution of the initial value problem

an(x)y(n)(x) + an-1(x)y(n-l)(x) + · · · + a1(x)yC^1 l(x) + ao(x)y(x) = 0

y(xo) = 0, yC^1 l(xo) = 0, ... , yCn-ll(xo) = 0