For a function $f(x,y)$ of two independent variables we have an incomplete specification of its partial derivatives as follows:
$\frac {\partial f(x,y)} {\partial x} = \frac {1} {g(x,y) \sqrt {1 - (\frac {k y} {x^{(1/3)}})^2}}$
$\frac {\partial f(x,y)} {\partial y} = \left(\frac {3 x} {4}\right)\left (\frac {k} {x^{(1/3)}}\right)^2 (2 y) \frac {1} {g(x,y) \sqrt {1 - (\frac {k y} {x^{(1/3)}})^2}}$
Problem: finding a suitable $g(x,y)$ that makes the partial derivatives converge to a single function $f(x,y)$ that fulfills the condition $f(x,0) = x$.
I will be grateful if people with many flight hours can offer suggestions for $g(x,y)$. Needless to say, I am not asking that they verify those suggestions, but in case someone would like, these are the inputs to Wolfram integrator:
1 / ( g(x,y as r) sqrt(1 - (k r / x^(1/3))^2) )
(3 t / 4) (k / t^(1/3))^2 (2 x) / ( g(x as t,y as x) sqrt(1 - (k x / t^(1/3))^2) )
Thanks in advance for your help.