1550078481-Ordinary_Differential_Equations__Roberts_

46 Ordinary Differential Equations

FUNDAMENTAL EXISTENCE AND UNIQUENESS THEOREM

Let R = {(x,y)i a< x < /3 and / < y < o} where a, /3, /,and o are

finite real constants. If f(x, y) is a continuous function of x and y in the

finite rectangle R , if fy(x, y) is a continuous function of x and y in R, and

if (c, d) ER, then there exists a unique solution to the initial value problem

(1) y' = J(x, y); y(c) =don some interval I= (c - h, c + h) where I is a

subinterval of the interval (a, /3).

This existence and uniqueness theorem tells us two things. First, when the

hypotheses are satisfied, there is a solution of the IVP (1) on some "small"

interval about c. And second , the solution is unique. That is, there is only

one solution of the IVP (1). Geometrically, the theorem states there is one

and only one solution of the differential equation in (1) which passes through

any point ( c, d) in the rectangle R.

Returning to the IVP ( 4) y' = 3y^213 ; y(O) = 0 and taking the partial

derivative of f(x, y) = 3y^213 with respect to y, we find fy(x, y) = 2y-^113.

Observe that fy(x, y) = 2y-^1 /^3 is defined and continuous for all x and for all

y i-0. So fy is continuous in any finite rectangle R which does not contain

any point of the x -axis (where y = 0). Thus, the fundamental existence and

uniqueness theorem just stated guarantees the existence of a -lmique solution

to the initial value problem

y' = 3y^213 ; y(c) = d

on some interval centered about c provided d -=f. 0. Notice that the IVP ( 4)

does not satisfy the hypotheses of the fundamental existence and uniqueness

theorem because d = 0.

The conditions (hypotheses) of the fundamental existence theorem are suffi-
cient to guarantee the existence of a solution to an initial value problem. And
the conditions of the fundamental existence and uniqueness theorem are suf-
ficient to guarantee the existence of a unique solution to an initial value prob-
lem. However, the conditions are not necessary conditions. Hence, if f(x, y)
does not satisfy the hypothesis of the fundamental existence theorem- that
is, if f(x, y) is not a continuous function of x and y in R, then we cannot
conclude that there is no solution to the initial value problem. If f(x, y) is
not continuous, the initial value problem may have a solution or it may not
have a solution. Likewise, if f(x, y) does not satisfy the hypotheses of the
fundamental existence and uniqueness theorem- that is, if f(x, y) is not a
continuous function of x and y or if fy(x, y) is not a continuous function of
x and y, then the initial value problem may have no so lution, it may have a
unique solut ion, or it may have multiple solutions.