I'm looking to a prove that a function that satisfies the following equations is actually $f(x,t)=x^3 \pm tx$ after changing coordinates.
Here are the equations:
1) $\frac{\partial^3 f}{\partial x^3}(0,0) \ne 0$ 2) $\frac{\partial^2 f}{\partial x \partial t}(0,0) \ne 0$ 3) $\frac{\partial^2 f}{\partial x^2}(0,0) = 0$ 4) $\frac{\partial f}{\partial x}(0,0) = 0$ 5) $\frac{\partial f}{\partial t}(0,0) = 0$
I've tried with Taylor's formula, and a few coordinates changes, but I haven't managed to go anywhere further than:
$f(x,t)=x^3+a*xt^2+b*t^3+c*t^2+d*xt$
Where $a, b, c$ and $d$ are smooth functions of $x$ and $t$. With $d(0,0) \ne 0$.
Any ideas? Or a book I could look at?
Thanks for the help