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I have

$$0 = F''(x) + p(x) F'(x) + cF(x)\\ p(x) = ab(1-x)$$

where $a$, $b$, $c$ are non-zero constants. I'm not very strong in the theories of 2nd order ODE, so I google'ed some solution methods. Turns out that there's a lot for constant coefficients, but many standard/introductory textbooks completely skip solution methods for the part with non-constant coefficients.

I am aware of the principle of superposition, but in order to use that, I first need to find two independent solutions. How can I proceed here (or in general?)

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1 Answer 1

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The solutions of this DE can be expressed in terms of Kummer functions:

$$F \left( x \right) ={ C_1}\,{{ M}\left(-{\frac {c}{2ab}},\, \frac12,\,\frac{ab}{2} \left( 1-x \right) ^{2}\right)}+{ C_2}\,{{ U} \left(-{\frac {c}{2ab}},\,\frac12,\,\frac{ab}{2} \left( 1-x \right) ^{2} \right)} $$

These are non-elementary special functions, which can be defined (surprise!) as solutions to second order linear differential equations...

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  • $\begingroup$ I suppose this holds for arbitrary real $C1, C2$. Could you elaborate how I could get that result? What are the steps to deriving this expression? $\endgroup$
    – FooBar
    May 12, 2015 at 15:53
  • $\begingroup$ And is there a simpler solution form when $c = ab$? $\endgroup$
    – FooBar
    May 12, 2015 at 16:11

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