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I am looking for the solution to the following system: $$ f_t(t,x) = -tx g(t,x), g_t(t,x) = (1-t)x f(t,x). $$

The equation comes from the integral equation $$ f(t,x)=1+ x \int_{0}^{1-t} (1-s)f(s,x)ds, $$ which is obtained from certain combinatorial generating function. Would you please give some comments on this?

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  • $\begingroup$ Unless there is a typo, this is an ODE with x as additional parameter. $\endgroup$ Oct 7, 2015 at 6:21
  • $\begingroup$ Yes, we can think it as a system of ODEs... $\endgroup$
    – hkju
    Oct 7, 2015 at 7:04
  • $\begingroup$ First : the $x$ parameter is irrelevant, it just multiplies the vector field so that it just gives a scaling in time. Second : what do you want to know ? $\endgroup$ Oct 7, 2015 at 7:59
  • $\begingroup$ Can we find the closed form or the explicit expression of the solution $f(t, x), g(t, x),$ not a series solution? $\endgroup$
    – hkju
    Oct 7, 2015 at 18:51

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