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Hello

I'm currently investigating the differential equation

$$f''=\left(\sum a_n x^n\right)f$$

I was wondering if anyone could please provide me with a lead as to where I may find the Green's functions? Sadly I have not been able to achieve the result through my own computations

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    $\begingroup$ What is a Green's function of a function? $\endgroup$
    – Andrey Rekalo
    Commented Feb 7, 2011 at 16:08
  • $\begingroup$ It's the Green's function of a differential equation. Sorry, I will edit that in :) $\endgroup$
    – backstoreality
    Commented Feb 7, 2011 at 17:01
  • $\begingroup$ This looks like it ought to yield to a standard application of Sturm-Liouville theory as soon as you make it clear what you want (for instance, what boundary conditions are you interested in?), and there are many e.g. "mathematical physics" textbooks which cover this subject. $\endgroup$
    – j.c.
    Commented Feb 7, 2011 at 22:17

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This is indeed fairly standard. Using the two linearly independent solutions to your ODE that you already have you can just follow the procedure outlined e.g. here to construct the Green function.

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  • $\begingroup$ Thank you for the link. The original notes that I had were apparently wrong! $\endgroup$
    – backstoreality
    Commented Feb 8, 2011 at 7:23
  • $\begingroup$ We're sorry. The site webspace.utexas.edu has been retired from service :( $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 21, 2021 at 17:52

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