1550078481-Ordinary_Differential_Equations__Roberts_

Introduction 21

Given a function y( x), we are usually able to determine where the function

is differentiable and to calculate its derivative. For example, the function

y(x) = x^3 -4x+l has derivative y' (x) = 3x^2 -4. Given the differential equation

y' ( x) = 3x^2 - 4, how do we recover the original function y( x) = x^3 - 4x + 1?
Integrating the differential equation y' ( x) = 3x^2 - 4 we get, not a single
function, but the one-parameter family of functions, y(x) = x^3 - 4x + C
where C is an arbitrary constant. In order to select the specific function
y(x) = x^3 - 4x + 1 from the family of fun~tions y(x) = x^3 - 4x + C, we must
specify some condition to be satisfied in addition to the differential equation.

The original curve y( x) = x^3 -4x+1 passes through the point ( 2, 1). Requiring

y(x) = x^3 - 4x + C to satisfy the condition y(2) = 1, we see that C must
satisfy the equation y(2) = 1 = 2^3 - 4(2) + C. Hence, C = 1 and we recover
the original function.

Proceeding one step further, we differentiate y( x) = x^3 - 4x + 1 twice and

find y" ( x) = 6x. Integrating this differential equation twice, we obtain the
two-parameter family of solutions y(x) = x^3 +Ax+ B where A and B are
arbitrary constants. To recover the original function, in this case, we must
specify two additional conditions which will require us to choose A = -4 and

B = l. For instance, we could require y(2) = 1 and y' (2) = 8, since the

original curve passes through the point (2, 1) and has slope 8 at (2, 1) - that
is ,

dyl =y'(2)=8.

dx x=2

Or, we could require y(l) = -2 and y(2) = 1 , since the original curve passes

through both (1, -2) and (2, 1).

The problem of solving the differential equation y" = 6x subj ect to the two

conditions

(9) y(2) = 1 and y'(2) = 8

is called an initial value problem and the conditions (9) are called initial
conditions.

The problem of solving the differential equation y" = 6x subject to the two

conditions

(10) y(l) = -2 and y(2) = 1

is called a boundary value problem and the conditions (10) are called
boundary conditions.

So two types of problems to be solved in the study of differential equations
are initial value problems and boundary value problems. A precise statement
of these two types of problems for n-th order ordinary differential equations
follows.