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- Feb 29, 2012

- 342

Consider the differential equation

$$x' = a_1 x + a_2 x^2 + a_3 x^3 + \cdots,$$

with $a_1 \neq 0$. Show that there exists a $C^2$ change of coordinates of the form $x = y + \alpha y^2$ that rewrites the equation (locally around $x=0$) as

$$y' = a_1 y + b_3 y^3 + \cdots,$$

that is, that eliminates the squared term.

I have no idea how to go about it.