1550078481-Ordinary_Differential_Equations__Roberts_

22 Ordin ary Differential Equations

DEFINITION Initia l Value Problem

The problem of finding a solution y = f(x ) of the n-th order differential

equation

(DE)

subject to a set of n conditions, called init ial conditio n s ,

(IC) y( xo) = co, y(ll (x o) = c1 ,... , yCn-l)(xo) = Cn-1

where x o, c 0 , c 1 , ... , Cn-l are real constants is called a n init ia l value prob-

le m (IV P ).

Notice that in a n initial value problem all n conditions t o b e satisfied a re
sp ecified a t a single value of the independent vari able-n a mely, at x 0.

DEFINITION Boundary Value Proble m

The problem of finding a solution y = f (x ) t o the differential equa tion

(DE) y(n) = F(x, y , y(l), ... , yCn-1))

subject to a set of n condit ions, called boundary conditio n s (BC) , which

sp ecify values of the function y or some of its derivatives a t two or more

distinct values of t he indep endent va ria ble is call ed a b oundary value

prob le m (B VP).

The theory associat ed with initial value problems is well est ablished and
relatively simple. As a consequence, throughout this t ext we will st a t e va rious
theorems which gua rantee the existence of solutions and theorems which gua r-
antee the uniqueness of solutions to different t ypes of initial value problems-
such as, initial value problems in which the differential equation is first-order ,
initial value problems in which t he different ia l equat ion is li near with const ant
coefficients, and initial value problems in which the differential equation is li n-
ear with varia ble coefficients. We will also define an an alogous init ial value
problem for syst em s of fir st -order differential equations and st at e an exist en ce
and uniqueness theorem for t his problem.