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Re: analytical ODE solution (a bit offtopic)
Posted:
Nov 29, 2012 9:31 AM


Leslaw Bieniasz <nbbienia@cyfkr.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



