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Dear all, I'm trying to find the general second-order ODE admitting

$x* = \alpha x$ $y* = \alpha^{k} y$

and reduce it to first-order plus a quadrature.

The general solution I found by using differential invariants is

y'' = \frac{1}{x} + H(y, - \frac{y'}{x})

where H is an arbitrary function.

But I have NO clue how to reduce it.

Please help me out here. Thank you in advance.

Edit: I made a big mistake here. Since the unit element of the scaling group is not 0, first I should transform into

$x* = e^{\epsilon} x$ $y* = e^{\epsilon \cdot k} y$ where $\alpha = e^{\epsilon}$.

Then the differential invariants are characterized by:

\frac{dx}{x}=\frac{dy}{ky}=\frac{d y'}{(k-1)y'} .

And the general solution is found as: y''=(k-1)\frac{y'}{x}-\frac{y'^{2}}{y}(k-x\frac{y'}{y})H(u,v) where $u(x,y) = \frac{x^{k}}{y}$ and v(x,y,y') = \frac{x^{k-1}}{y'}

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    @Sam Lisi: As I edited above, I made an egregious mistake. Sorry for that :(2011-04-05

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