UnkleRhaukus
  • UnkleRhaukus
\[\frac{\mathrm dx}{\mathrm dt} = -4\pi^2x+x^2\] \[x_0 = \pi^2,\qquad v_0=0\]
Differential Equations
katieb
  • katieb
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Empty
  • Empty
Is \(\pi\) the circle constant?
UnkleRhaukus
  • UnkleRhaukus
yes
Empty
  • Empty
I guess it appears to be a separable differential equation, unless there's some trick going on haha. This is kind of odd looking.

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UnkleRhaukus
  • UnkleRhaukus
numeric methods
Empty
  • Empty
It has a general solution \[x=\frac{4 \pi^2}{e^{4\pi^2 (c+t)}+1}\]
Astrophysics
  • Astrophysics
Yeah, but how did you get that, I thought separable as well, but what is the numeric method :O
Empty
  • Empty
I don't know, there are many numerical methods
UnkleRhaukus
  • UnkleRhaukus
the questing is asking me to use the leapfrog integration method, but i'm not convinced it is stable
UsukiDoll
  • UsukiDoll
@Astrophysics there was an x left over... usually when it's separable it would be in the form of h(y) dy = f(x) dx and then integrate both sides but that's not the case in this equation :/
Astrophysics
  • Astrophysics
Ah ok, and I see leap frog method, is this a mechanics problem..mhm
Empty
  • Empty
Yeah, that's how I got my answer, partial fractions. This is separable @UsukiDoll
Astrophysics
  • Astrophysics
Wolfram confirms, I don't really know leapfrog integration very well, this does remind me of a simple harmonic oscillator, @Empty you know how to do it using the method
Empty
  • Empty
Ok but the real question is, "How do I show that leapfrog integration is stable" and that I don't know.
UsukiDoll
  • UsukiDoll
X( completely missed that... yes it is separable... (after hours...mind can't think). But what is leapfrog integration?
UsukiDoll
  • UsukiDoll
\[\frac{\mathrm dx}{\mathrm dt} = -4\pi^2x+x^2 \] \[\frac{dx}{-4 \pi^2x+x^2} = dt\] \[\frac{dx}{x(-4 \pi^2+x)} = dt\] and then partial fractions... (not doing the rest)
Astrophysics
  • Astrophysics
Lol I don't think he's allowed to use that
UsukiDoll
  • UsukiDoll
-_- k forget it *tosses it in the trash*
UsukiDoll
  • UsukiDoll
was fun when it was legit.
Astrophysics
  • Astrophysics
Haha have a medal, @Michele_Laino may be able to help you
UnkleRhaukus
  • UnkleRhaukus
Catastrophic, error sorrys
UsukiDoll
  • UsukiDoll
nah... could be something I haven't done or idk... it's similar to asking me to do a PDE involving Laplace Equation at after 3 am. My brain can't function at this hour..

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