1549301742-The_Theory_of_Difference_Schemes__Samarskii

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whose roots are

c ± J c^2 - 4 ab

2b

Preliminaries

Three possibilities of interest in accordance with the discriminant sign are:

1) If D = c^2 - 4ab > 0, then the preceding equation has two distinct real

roots

c+VJ5

2b

c-VJ5

2b

Two different particular solutions y~^1 ) = q~ and y~^2 ) = q~ corresponding to

q 1 and q 2 are linearly independent, since

q;· q~'

q~+l q;+i

and constitute what is called the general solution of ( 45):

Yk = C I ql k + c 2 q2 k

where C 1 and C 2 are arbitrary constants.

2) If D = c^2 - 4ab = 0, then q 1 = q 2 = c/(2b) = q 0 and y~l) = q~ and

y1^2 l = k q~ can be declared to be linearly independent particular solutions.

Indeed, substituting yf^2 l = k q~ into (45) we obtain by minor changes

by1~ 1 -cy1

2

l+ay1

2

] 1 = [b(k+l)q;-ckq 0 +a(k-l)] q~-I

=q~-I k(bq;-cq 0 +a)+q~-I (bq~-a)=q~-I (bq~-a)=O,

since bq;-a= b[c/(2b)]2-a= D/(4b) = 0. The discriminant

q~ k q~

q~+l (k + 1) q~+I

assures us that q~ and k q; are linearly independent and can be chosen as
a basis for constructing the general solution of ( 45):