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Particular Integral where f(x) is a negative order polynomial?



What form of approximation do I use for the forcing of a second order inhomogeneous D.E which is a polynomial of a negative degree in order to find it's particular integral? I know for a polynomial of degree n where n is an integer, y would just be another polynomial of degree n.

Any ideas?
Reply 1
Original post by nugiboy


What form of approximation do I use for the forcing of a second order inhomogeneous D.E which is a polynomial of a negative degree in order to find it's particular integral? I know for a polynomial of degree n where n is an integer, y would just be another polynomial of degree n.

Any ideas?


Just looking at it I'd be tempted to try y = A/x^2 - I think the derivatives will knock the power down by 1 each time but the multiplying factors will bring everything back to a multiple of 1/x^2.
Reply 2
Original post by davros
Just looking at it I'd be tempted to try y = A/x^2 - I think the derivatives will knock the power down by 1 each time but the multiplying factors will bring everything back to a multiple of 1/x^2.


Hmm that doesn't seem to work..
Reply 3
since the homogeneous equation is Cauchy-euler, why not try a multiple of part of the solution you`d get if you had one repeated root?

namely: Aln(x)x2\displaystyle \frac{Aln(x)}{x^2} ?

(I`ll leave you to figure out why - hint: 2nd order, exact - try method of undetermined coeffs)
(edited 10 years ago)

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