z(k, e) = 4*ak+e z(k, l) = 4-'ak+. or

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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Example B
The equation
z(k +1, e) = 42(k, l+ 1)
(5.84)
leads to the conclusion that A
4µ. Therefore, equation (5.84) has the special
solutions
z(k, l) = 4*µk+e.
(5.85)
Summing expressions of this form allows the conclusion
z(k, l) = 4* f (k + e),
(5.86)
where f is an arbitrary function of k + l.
The separation-of-variables method, where zp(k, l) = CkDe, gives
Ck+1De = 4CkDe+1,
(5.87)
which can be written in either of the forms
4De+1
Ck+1
4Ck
De+1
= a
De
Ck+1
or
= a.
(5.88)
Ck
De
The respective solutions to these equations give
z(k, l) = 4*ak+e
or 2(k, l) = 4-la*+e.
(5.89)
Summing over a allows us to obtain the general solution
z(k, e) = 4* f (k + l) or z(k,l) = 4"g(k, l),
(5.90)
where f and g are arbitrary functions of k + l.
Transcribed Image Text:Example B The equation z(k +1, e) = 42(k, l+ 1) (5.84) leads to the conclusion that A 4µ. Therefore, equation (5.84) has the special solutions z(k, l) = 4*µk+e. (5.85) Summing expressions of this form allows the conclusion z(k, l) = 4* f (k + e), (5.86) where f is an arbitrary function of k + l. The separation-of-variables method, where zp(k, l) = CkDe, gives Ck+1De = 4CkDe+1, (5.87) which can be written in either of the forms 4De+1 Ck+1 4Ck De+1 = a De Ck+1 or = a. (5.88) Ck De The respective solutions to these equations give z(k, l) = 4*ak+e or 2(k, l) = 4-la*+e. (5.89) Summing over a allows us to obtain the general solution z(k, e) = 4* f (k + l) or z(k,l) = 4"g(k, l), (5.90) where f and g are arbitrary functions of k + l.
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