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From: master1729 on 26 Apr 2010 12:29
slawek wrote :

> I am looking for solutions of this equation:
>
> y'[t] = y[t] - a y[t] y[t-1] y[t-2]
>
> It would be assumed that y[t] =0 for t < 0 .

i wouldnt assume that if i were you.

id say the equation holds only for t > 2.

then we get :

log(y[t]) = C - a y[t-1]y[t-2]

thus we prefer t > 2 and y[t] =/= 0.

for large t , assuming lim y[t-2]/y[t-1] = 1.

we arrive at the approximation

sqrt(-(log(y[t]) - C)/a) = y[t-1]

y[t] = exp(C - a y[t-1]^2)

thus we get locally the superfunction :

y[t] = superfunction of C_0 * exp(-a x^2)

a gaussian superfunction so to say.

well locally for large t at least.

that might explain why your pic goes to 0 so fast.

that might also explain why you wont find many elementary solutions apart from the exponential ones.

the exponential ones are those who dont have lim y[t-2]/y[t-1] = 1 of course.

im sure you can find the exponential solutions by yourself. ( use dummy variables )

btw, didnt i post that equation once , in relation to tetration ?

>
> TIA
> slawek
>

regards

tommy1729

the master
From: slawek on 26 Apr 2010 17:07

Użytkownik "master1729" <tommy1729(a)gmail.com> napisał w wiadomości grup
dyskusyjnych:372845407.25941.1272313817520.JavaMail.root(a)gallium.mathforum.org...
> that might also explain why you wont find many elementary solutions apart
> from the exponential ones.

LOL

> btw, didnt i post that equation once , in relation to tetration ?

I don't know, I am not your nurse.



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