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Re: analytical ODE solution (a bit off-topic)
Posted:
Nov 29, 2012 9:31 AM
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Leslaw Bieniasz <nbbienia@cyf-kr.edu.pl> writes:
>
>
>Hi,
>
>I need to solve analytically a certain second order ODE,
>which takes the general form
>
>y''(x) - p(z,x)*y(x) = 0.
>
>where p(z,x) is a polynomial and y(x) is to be determined. The polynomial
>depends on a complex parameter z.
>
>The problem is that I need a possibly approximate but analytical solution,
>perhaps in the form of some truncated series (but not the series in powers
>of x), not just numerical values of the solution.
>
>Are there any techniques available? Some kind of
>successive approximations that really work (are convergent)?
>Someone has suggested to me the WKB approximation, but I think
>it is not suitable, because the above equation is not singularly
>perturbed.
>
>Leslaw
>
>
>
maybe Kamke helps:
''Differentialgleichungen, Loesungsmethoden und Loesungen''
shows 2.31 how to reduce this to a Riccati DGL with inhomogenity
p(x,z) and in A4.9 how to deal with a general Riccati DGL.
But since most Riccati -DGL cannot be solved by elementary functions
or quadrature, my hope is not that large. Maybe a series solution,
possibly done piecewise, helps.
hth
peter
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