Fuo = a, (24) %3D (be – cd ) uz + (bg - df ) uz + (bs - dr ) us Fur (25) (d u1 +e uz + g u3 + su4)? (be – cd ) u1 + (cg -ef ) uz + (cs - er ) us Fuz (26) (d u1 +e uz + g u3 + su4) (bg – df ) u1 - (cg - ef ) uz + (fs - gr) u4 (d u1 +e uz +g uz + sus) Fu, (27) and (bs - dr ) u1- (cs - er ) u2 - (fs - gr) u3 (d uj +e uz +9 u3 + su4)? Hence, the proof follows by some computations and it will be omitted here. Fus (28) bui + cuz + fu3 + rus F(uo, ..., u4) = auo + (8) dui + euz + guz + su4

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Explain the determine purple and the equation 8 is here

Lemma 7 For uny values of the quotient , and ;, the function F(uo,
.., u4) defined by Eq.(8) is momotonic in its aryuments.
Proof: By differentiating the function F(uo, ..., u4) given by the formula (8)
with respect to u (i = 0, 1, 2, 3, 4) we obtain
Fuo = a,
(24)
(be – cd ) u2 + (bg - df ) uz + (bs - dr ) u4
(d u1 +e uz + g u3 + su4)?
(be – cd ) u1 + (cg - ef) uz + (cs
Fu
(25)
%3D
- er ) u4
Fuz =
(26)
(d u1 +e uz +g uz + su4)
(bg – df ) u1 - (cg - ef ) u2 + (fs -gr) u4
(d u1 +e u2 +g u3 + su4)
Fu, =
(27)
and
- (bs - dr ) u1 - (cs - er ) u2 - (fs -gr) u3
(d u1 +e u2 +g u3 + su4)
Hence, the proof follows by some computations and it will be omitted here.
Fu
(28)
bui + cu2 + fuz + ru4
F(uo, ..., us) = auo +
(8)
duj + euz + guz + su4
Therefore it follows that
8F(uo,..u4) = a,
Transcribed Image Text:Lemma 7 For uny values of the quotient , and ;, the function F(uo, .., u4) defined by Eq.(8) is momotonic in its aryuments. Proof: By differentiating the function F(uo, ..., u4) given by the formula (8) with respect to u (i = 0, 1, 2, 3, 4) we obtain Fuo = a, (24) (be – cd ) u2 + (bg - df ) uz + (bs - dr ) u4 (d u1 +e uz + g u3 + su4)? (be – cd ) u1 + (cg - ef) uz + (cs Fu (25) %3D - er ) u4 Fuz = (26) (d u1 +e uz +g uz + su4) (bg – df ) u1 - (cg - ef ) u2 + (fs -gr) u4 (d u1 +e u2 +g u3 + su4) Fu, = (27) and - (bs - dr ) u1 - (cs - er ) u2 - (fs -gr) u3 (d u1 +e u2 +g u3 + su4) Hence, the proof follows by some computations and it will be omitted here. Fu (28) bui + cu2 + fuz + ru4 F(uo, ..., us) = auo + (8) duj + euz + guz + su4 Therefore it follows that 8F(uo,..u4) = a,
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