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?