How can I get an asymptotic solution for 2nd order differential equation.

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For the 2nd order differenttiaal equation:
where α and are constants.
How can I get an asymptotic solution, where for z=0. Thanks for the help.
  11 个评论
Dereje
Dereje 2019-2-1
@David I see also in some papers similar solution like you stated above (). Can you please show me the steps how to come up with this solution.
David Goodmanson
David Goodmanson 2019-2-1
编辑:David Goodmanson 2019-2-1
HI Dereje,
There seems to be inconsistency about M^2 vs. M, so it's preferable to quit M and use Torsten's notation for a starting point:
u'' = c u^(1/4)
LIke a lot of differential equations you assume a solution and verify that it works. (It was kind of exasperating sometimes in the diffyq course, when the instructor would put up some differential equation, pull a hypothetical solution out of the air and say, let's see if this works. Which of course it always did). Anyway, assume u = a z^n, plug that in, get
n(n-1) z^(n-2) = c (a z^n)^(1/4) = c a^(1/4) z^(n/4)
Equate powers of z on each side, find out n = 8/3. Equate coefficients in front on each side and you can solve for a. You get one solution for positive z that way, and you can see that in trying to find others, expansion in powers of z^(8/3) makes some sense

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