Problem 1270 (difficulty: 7/10)

Assume that \(\displaystyle f:\R^2\to\R\) is differentiable and for all \(\displaystyle x,y\) we have

\(\displaystyle y^2\cdot D_1f(x,y)=x^2\cdot D_2f(x,y). \)

Prove that \(\displaystyle f(x,y)=g(x^3+y^3)\) for some \(\displaystyle g\). Is it necessarily true that the function \(\displaystyle g\) is differentiable at \(\displaystyle 0\)?

Hint-->

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