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I need help to verify a solution to a partial differential equation. Here is the solution:

$$f_t(s)=2 \sqrt{\frac{t}{s}}K_1(2\sqrt{t s})$$

where $K_1$ is a modified Bessel function of the second kind.

Here's the differential equation:

$$ tf_{ttt}=sf_s$$

I did computations with Bessel identities and then used Wolfram Alpha to verify the result but I am not confident that it's correct.

Does $f_t(s)$ satisfy that equation?

Thank you!

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    $\begingroup$ You have $f_t(s)$ in the 1st equation, and $f_s$ in the second one. Do I understand correctly that $f_t(s)=\partial_t f(s, t)$ and $f_s=\partial_s f(s, t)$? $\endgroup$
    – yarchik
    Jul 14, 2023 at 14:19
  • $\begingroup$ What exactly $f_{ttt}$ means? Is it $\partial^3/\partial t^3 f(s, t)$? $\endgroup$
    – yarchik
    Jul 14, 2023 at 14:21
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    $\begingroup$ Why not writing everything properly and even attaching a Mathematica code to the question? It is really annoying that one has to guess. $\endgroup$
    – yarchik
    Jul 14, 2023 at 14:22
  • $\begingroup$ What is modified Bessel function of the second kind? BesselK[1, #] &? $\endgroup$
    – cvgmt
    Jul 14, 2023 at 14:23
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    $\begingroup$ People here generally like users to post code as copyable Mathematica code as well as images or TeX, so they can copy-paste it. It makes it convenient for them and more likely you will get someone to help you. You may find the meta Q&A, How to copy code from Mathematica so it looks good on this site, helpful $\endgroup$
    – Michael E2
    Jul 14, 2023 at 14:31

1 Answer 1

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FullSimplify[t^2*D[f[t, s], {t, 3}] == s^2*D[f[t, s], s] /. 
f -> Function[{t, s}, 2*Sqrt[t/s]*BesselK[1, 2*Sqrt[t*s]]]]

(*True*)

Looks like your equation is: $$t^2 \frac{\partial ^3f(t,s)}{\partial t^3}=s^2 \frac{\partial f(t,s)}{\partial s}$$

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