Question

\[\frac{\mathrm dx}{\mathrm dt} = -4\pi^2x+x^2\]
\[x_0 = \pi^2,\qquad v_0=0\]

Answer 1

Is \(\pi\) the circle constant?

Answer 2

yes

Answer 3

I guess it appears to be a separable differential equation, unless there's some trick going on haha. This is kind of odd looking.

Answer 4

numeric methods

Answer 5

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

Answer 6

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

Answer 7

I don't know, there are many numerical methods

Answer 8

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

Answer 9

@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 :/

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

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

Answer 14

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

Answer 15

\[\frac{\mathrm dx}{\mathrm dt} = -4\pi^2x+x^2 \]
\[\frac{dx}{-4 \pi^2x+x^2} = dt\]
\[\frac{dx}{x(-4 \pi^2+x)} = dt\]
and then partial fractions... (not doing the rest)

Answer 16

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

Answer 17

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

Answer 18

was fun when it was legit.

Answer 19

Haha have a medal,


@Michele_Laino may be able to help you

Answer 20

Catastrophic, error
sorrys

Answer 21

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..