How can I solve this delay differential equation?

Asked May 10 '21 at 14:23

Active May 10 '21 at 15:23

Viewed 114 times

I'm solving a problem where I encountered a DDE: $$ y'(t)=r\left(1-\frac{y(t-q)}{C}\right)y(t)$$ Where $r$, $q$ and $C$ are real constants.

Now, with $q=0$ it's pretty easy and the solution is just a logistic function. But what if $q\neq0$? I can easily write a program that numerically solves this, but is there a method to actually find $y(t)$?

edited May 10 '21 at 15:23

callculus42

30,550

asked May 10 '21 at 14:23

Francesco Sollazzi

Does that y in the y(t-q) term represent y(t) itself? – Ritam_Dasgupta May 10 '21 at 14:31
Yes, it's $y(t)$ – Francesco Sollazzi May 10 '21 at 14:31
3

Since you can rescale $t$, without loss of generality you can assume $q = 1$. Still, delay differential equations are difficult to do analytically. – eyeballfrog May 10 '21 at 14:33
@eyeballfrog oh, thanks for the link, I'll definitely check it out. – Francesco Sollazzi May 10 '21 at 14:35
Why the downvote? Can I improve the question? – Francesco Sollazzi May 10 '21 at 14:37
1

We can also get this to a single delay parameter with the substitutions $y(t) = Cu(rt)$, $s = rt$ and $a = qr$, giving $u'(s) =u(s)\left[1 -u(s-a)\right]$. A different rescaling of $t$ can also give $u'(s) = a u(s) \left[ 1- u(s - 1)\right]$. – eyeballfrog May 10 '21 at 14:45
Being a delay differential equation, it's far from "ordinary". – Robert Israel May 10 '21 at 14:50
@RobertIsrael I didn't know DDEs, edited. – Francesco Sollazzi May 10 '21 at 14:58

How can I solve this delay differential equation?

0 Answers0