1550078481-Ordinary_Differential_Equations__Roberts_

Introduction 11

1

So the function y = x + - is a solution of x^2 y" + xy^1 -y = 0 on ( -oo, 0),

x

on (0, oo ), and on any subinterval of these intervals, but y = x + .£ is

x

not a solution of the differential equation on any interval which includes

the point x = 0.

In calculus, you learned the definition of the derivative of a function, and you
calculated the derivative of a few functions from the definition. Also, you may
have seen the definition of the left-hand derivative and right-hand derivative of
a function in calculus. Since we want to determine where piecewise functions
are differentiable, we provide those definitions again.

DEFINITION Derivative, Left-hand Derivative,

and Right-hand Derivative

Let f(x) be a real valued function of a real variable defined on some

interval I and let c be in the interval I.

The derivative of f at c is

f^1 ( c )-- l· Jill f(c+h)-f(c) l '

h->O 1

provided the limit exists.

The left-hand derivative of f at c is

f^1

( ) _

1

. f(c + h) - f(c)

• c - h->OJill - h ' provided the limit exists.

The right-hand derivative of f at c is

f^1

( ) _

1

. f(c + h) - f(c)

+ c - h->O+ Jill h ' provided the limit exists.

REMARKS: The derivative of the function f at c exists if and only if

both the left-hand derivative at c and the right-hand derivative at c exist and

are equal. Recall from calculus, if f is differentiable at c, then f is continuous

at c. In order to show that a function f is not differentiable at c, we can

show that (1) the function f is not continuous at c, or (2) either the left-hand

derivative at c or the right-hand derivative at c does not exit, or (3) the left-
hand derivative at c and the right-hand derivative at c both exist, but they
are not equal.