2
$\begingroup$

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?)

$\endgroup$
3
$\begingroup$

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...

$\endgroup$
  • $\begingroup$ 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 '15 at 15:53
  • $\begingroup$ And is there a simpler solution form when $c = ab$? $\endgroup$ – FooBar May 12 '15 at 16:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.