1550078481-Ordinary_Differential_Equations__Roberts_

342

and

2-.A

4

1

3

-1-.A

2

Ordinary Differential Equations

-1

0

3-.A (

2-.A

=det ~

3

-1-.A

2

-1 )

3~.A

= (2 - .A)(-1-.A)(3-.A)+ (3)(0)(1) + (-1)(4)(2)

- (2 - .A)(0)(2) - (3)(4)(3 - .A) - (-1)(-1 - .A)(l)

= (2 - .A)(-1-.A)(3-.A)+ 0-8 - 0-12(3-.A)+ (-1-.A)

= -.A^3 + 4.A^2 - .A - 6 -8 - 36 + 12.A - 1 - .A

= -.A^3 + 4.A^2 + 10.A - 51.

The set of n simultaneous linear equations

(2)

in then unknowns x 1 , x 2 , ... , Xn may be rewritten in matrix notation as

(2') Ax=b

where A is the n x n matrix

(""

a12

"'")

a21 a22

..•. :I~

A=

an1 an2

and x and b are the n x 1 column vectors

If b = 0 , then systems (2) and (2') are said to be homogeneous; otherwise,
they are called nonhomogeneous.