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Aug
28
reviewed Approve Limit of a partial sum
Aug
25
comment Solve the PDE: $u_{xx} - 3u_{xt} - 4u_{tt} = 0$
-1. The answer is incorrect.
Aug
24
comment Solve the PDE: $u_{xx} - 3u_{xt} - 4u_{tt} = 0$
I am really confused how you got from $u_{\eta\xi}=0\implies u=c_1\eta+c_2\xi+c_3$, whereas the correct answer should be $u(\eta,\xi)=C_1(\eta)+C_2(\xi)$.
Aug
18
comment Reference Request: Group Theory via the Group Action Perspective
@BhaskarVashishth For an opposite direction, please see Arnold, V.I., ODE text. :)
Aug
11
reviewed Approve Limit of $(n!)^{1/n}$ as $n\rightarrow \infty$.
Aug
5
reviewed Approve Is a minimum of RMSE also a minimum of MAE?
Aug
1
reviewed Approve why does following series diverges
Aug
1
reviewed Approve In an inner product space, if the matrix is symmetric, is an eigenspace necessarily orthogonal to the range space?
Aug
1
reviewed Approve Help With Finding A Basis
Aug
1
reviewed Approve Are there an infinite number of open balls in an open set in a metric space?
Jul
30
comment Periodic orbits of “even” perturbations of the differential system $x'=-y$, $y'=x$
@Did Thanks for the bounty. Actually I was wrong. In Andronov et al. Theory of Bifurcations of Dynamic Systems on a Plane (which was translated into English, but I am not sure it is easy to find) there is an elementary proof why the first index must be odd to be nonzero (in Section 24).
Jul
28
comment Periodic orbits of “even” perturbations of the differential system $x'=-y$, $y'=x$
@Did. You are welcome. Actually, for $f(u)=g(u)=u^2$ the answer is the center, hence all $\alpha_j=0$. It can be proved by first rotating the coordinate system $\pi/4$ and then notice that the transformation $(t,y)\mapsto(-t,-y)$ (for the new coordinate $y$) leaves the system invariant. It looks like exactly the same argument works for arbitrary case $f=g$ and both are even.
Jul
28
awarded  Revival
Jul
28
revised Periodic orbits of “even” perturbations of the differential system $x'=-y$, $y'=x$
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Jul
28
revised Periodic orbits of “even” perturbations of the differential system $x'=-y$, $y'=x$
added 452 characters in body
Jul
28
answered Periodic orbits of “even” perturbations of the differential system $x'=-y$, $y'=x$
Jul
28
comment Periodic orbits of “even” perturbations of the differential system $x'=-y$, $y'=x$
@Did Honestly, I have no idea why nonzero $\alpha$ must have an odd index, I just saw an attribution to Lyapunov, who proved it. I tried to find a proof in one of the compendium volumes by Andronov et al., but failed. $\alpha_3$ can be calculated (up to a scaling factor) as $f'''(0)+g'''(0)$, which is always zero for your $f$ and $g$. I will add how to compute $\alpha$-s in a separate answer, it requires some tedieous (but easy with a computer algebra system) calculations.
Jul
28
answered Periodic orbits of “even” perturbations of the differential system $x'=-y$, $y'=x$
Jul
28
reviewed Approve Why is it so hard to find a generating function for Somos' sequence?
Jul
21
revised How do I demonstrate that the given functions solve this system of ODEs?
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