1550078481-Ordinary_Differential_Equations__Roberts_

318 Ordinary Differential Equations

find the general solut ion of linear systems of first-order equations with con-

stant coefficients. (System (8) is an example of a system of this type.) And

in chapter 9, we will solve several appli cations which involve linear systems

with constant coefficients.

DEFINITION An Initial Value Problem for a System of

Differential Equations

An initial value problem for a system of first-order differential equa-

tions consists of solving a system of equations of the form (1) subject to a

set of constraints, called initial conditions (IC), of the form y1(c) =di,

Y2(c) = d2, ... , Yn(c) = dn.

For example, the problem of finding a solution to the system

(8)

Y~ = Y2 = Ji (x, Y1, Y2)

Y~ = -Y1 = f2(x, Y1, Y2)

subject to the initial conditions

(9) Y1(0) = 2, Y2(0) = 3

is an initial value problem. The so lution of the initial value problem (8)-(9)

on the interval (-oo, oo) is

{Y1(x) = 3sinx + 2cosx, y2(x) = 3cosx - 2sinx}.

Verify this fact by showing that y 1 ( x) and Y2 ( x) satisfy the system ( 8) on the

interval ( -oo, oo) and the initial conditions (9).

Notice that in an initial value problem all n conditions to be satisfied are

specified at a single value of the independent variable-the value we have

called c. The problem of solving the system (8) subject to the constraints

(10) Y1(0) = 2, Y2(7r) = 3

is an example of a boundary value problem. Observe that the constraints for

Y1 and Y2 are specified at two different values of the independent variable-

namely, 0 and 7r. Verify that the set

{yi ( x ) = -3 sin x + 2 cos x, Y2 ( x ) = -3 cos x - 2 sin x}

is a solution of the boundary value problem (8)-(10) on the interval (-oo, oo).