1550078481-Ordinary_Differential_Equations__Roberts_

366 Ordinary Differential Equations

subject to the n constraints

(4b)

If we let y(x), b(x), and d be then x 1 column vectors

Y2(x)

(

Y1 (x))

y(x) = : '

Yn(x)

and if we let A(x) be then x n matrix

a11(x) a12(x) · · · a1n(x)

an1(x) an2(x) · · · ann(x)

then using matrix notation we can write the linear first-order system initial

value problem (4) more concisely as

(5) y' = A(x)y + b(x); y(c) = d.

For example, using matrix-vector notation the system of equations

y~ = 3y1 - 4y2 + x

y~ = -2y1 + Y2 - sinx

can be written as

(y~) y~ = ( -^3 2 -4 1 ) (Yi) Y2 + ( -sin x x ) ·

DEFINITIONS Homogeneous and Nonhomogeneous

Linear Systems

The system of linear first-order differential equations

(6) y' = A(x)y + b(x)

is said to be homogeneous provided b(x) = 0 and nonhomogeneous

provided b(x)-=/= 0.

Thus, a homogeneous linear system is one of the form

(7) y' = A(x)y.