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

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Dereje 2019-1-29

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https://ww2.mathworks.cn/matlabcentral/answers/442032-how-can-i-get-an-asymptotic-solution-for-2nd-order-differential-equation

编辑： David Goodmanson 2019-2-1

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.

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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 2019-2-1

编辑：David Goodmanson 2019-2-1

在 MATLAB Online 中打开

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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