szklanovqq

2022-11-19

Completing partial derivatives to make them converge
For a function $f\left(x,y\right)$ of two independent variables we have an incomplete specification of its partial derivatives as follows:
$\frac{\mathrm{\partial }f\left(x,y\right)}{\mathrm{\partial }x}=\frac{1}{g\left(x,y\right)\sqrt{1-\left(\frac{ky}{{x}^{\left(1/3\right)}}{\right)}^{2}}}$
$\frac{\mathrm{\partial }f\left(x,y\right)}{\mathrm{\partial }y}=\left(\frac{3x}{4}\right){\left(\frac{k}{{x}^{\left(1/3\right)}}\right)}^{2}\left(2y\right)\frac{1}{g\left(x,y\right)\sqrt{1-\left(\frac{ky}{{x}^{\left(1/3\right)}}{\right)}^{2}}}$
Problem: finding a suitable $g\left(x,y\right)$ that makes the partial derivatives converge to a single function $f\left(x,y\right)$ that fulfills the condition $f\left(x,0\right)=x$
I will be grateful if people with many flight hours can offer suggestions for $g\left(x,y\right)$. 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:

h2a2l1i2morz

Hint: From your data one obtains
$-\frac{{f}_{x}}{{f}_{y}}=\frac{-2}{3{k}^{2}y{x}^{1/3}}.$
It follows that the curves $f\left(x,y\right)=$const. satisfy the separable differential equation
${y}^{\prime }=\frac{-2}{3{k}^{2}y{x}^{1/3}}.$

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