1550078481-Ordinary_Differential_Equations__Roberts_

Introduction 9

4. Functions of y or any of its derivatives such as eY or cosy' cannot
   appear in the equation.

Equations (1), (2), and (3) are all nonlinear ordinary differential equations.

Equation (1) is nonlinear because of the term cosy. Equation (2) is nonlinear

because of the term (dy/dx )^2. And equation (3) is nonlinear because of the

terms ( d^4 y / dt^4 )^3 and ty( d^2 y / dt^2 ). The equations

(8) yC^2 l - 3y(l) + 2y = x

(9)

a re both linear ordinary differential equations.

DEFINITION Solution of a Differential Equation

A solution of the n-th order ordinary differential equation

F(x, y(l), ... , y(n)) = 0

on an interval I = (a, b) is a function y = f(x) which is defined on I ,

which is at least n times differentiable on I, and which satisfies the equation

F(x, j(l), ... , f(n)) = 0 for all x in I.

Since a solution y = f(x) is at least n times differentiable on t h e interval I ,

the functions f(x), jC^1 l(x), ... , jCn-^1 l(x) are all continuous on I. Usually

the interval I is not specified explicitly, but it is understood to be the largest

possible interval on which y = f(x) is a solution.

Let p(x) be a polynomial. By definition, a solution of the equation p(x) = 0

is a root of the polynomial. To determine if any particular complex number
s is a root of p(x) or not, we compute p(s) and see if the value is 0 or not.
That is, to determine if a number s is a root of a polynomial, we do not need
to be able to "solve the polynomial" by factoring, by using some formula such
as the quadratic formula, or by using a numerical technique such as Newton's
method to find the roots of the polynomial. All we need to do is substitute the

numbers into the polynomial p(x) and see if p(s) = 0 or not. Similarly, if we

write a differential equation so that 0 is the right-hand side of the equation,

then to determine if some function y(x) is a solution of the differential equation

or not, all we need do is (1) determine the order n of the given differential

equation, (2) differentiate t he function y(x) n times, (3) substitute y(x) and

its n derivatives into t he given differential, and ( 4) see if the result is 0 or

not. If the result is not 0 , then the function is not a solution of the given

differential equation. If the result is 0 , then the function y(x) is a solution on

any interval on which it and its n derivatives are simultaneo usly defined.