Verification of solution to differential equation

Question

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!

Answer 1

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: $$t^2 \frac{\partial ^3f(t,s)}{\partial t^3}=s^2 \frac{\partial f(t,s)}{\partial s}$$