## UnkleRhaukus one year ago $\frac{\mathrm dx}{\mathrm dt} = -4\pi^2x+x^2$ $x_0 = \pi^2,\qquad v_0=0$

1. Empty

Is $$\pi$$ the circle constant?

2. UnkleRhaukus

yes

3. 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.

4. UnkleRhaukus

numeric methods

5. Empty

It has a general solution $x=\frac{4 \pi^2}{e^{4\pi^2 (c+t)}+1}$

6. Astrophysics

Yeah, but how did you get that, I thought separable as well, but what is the numeric method :O

7. Empty

I don't know, there are many numerical methods

8. UnkleRhaukus

the questing is asking me to use the leapfrog integration method, but i'm not convinced it is stable

9. 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 :/

10. Astrophysics

Ah ok, and I see leap frog method, is this a mechanics problem..mhm

11. Empty

Yeah, that's how I got my answer, partial fractions. This is separable @UsukiDoll

12. 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

13. Empty

Ok but the real question is, "How do I show that leapfrog integration is stable" and that I don't know.

14. UsukiDoll

X( completely missed that... yes it is separable... (after hours...mind can't think). But what is leapfrog integration?

15. 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)

16. Astrophysics

Lol I don't think he's allowed to use that

17. UsukiDoll

-_- k forget it *tosses it in the trash*

18. UsukiDoll

was fun when it was legit.

19. Astrophysics

20. UnkleRhaukus

Catastrophic, error sorrys

21. 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..