Rewriting a ODE in -t

**Dinkydoe** · September 30th 2010, 04:47 AM

It feels like a stupid question...

given the system:
$\frac{dx}{dt} = y$
$\frac{dy}{dt} = Ay-x(1-x)$

with x=x(t), y=y(t)

How would I rewrite the system in, say $\hat{t}=-t$ ?

**Ackbeet** · September 30th 2010, 05:18 AM

Well, you have an autonomous system there, meaning that you don't have to worry about the RHS's so much. Try this:

$\displaystyle{\frac{dy}{d\hat{t}}=\frac{dy}{dt}\,\ frac{dt}{d\hat{t}}.}$

**Dinkydoe** · September 30th 2010, 08:10 AM

Doesn't this just give... $\frac{dy}{d\hat{t}} = -\frac{dy}{dt}$

Don't we also want to have $y(\hat{t})$ in our equations?

How do we deal with that?

**Ackbeet** · September 30th 2010, 08:14 AM

Can't you just make the substitution outright? I'm thinking

$\displaystyle{-\frac{dy}{d\hat{t}}=y(\hat{t})}$

$\displaystyle{-\frac{dy}{d\hat{t}} = Ay(\hat{t})-x(\hat{t})(1-x(\hat{t}))}.$

Wouldn't that do the job?

**Dinkydoe** · September 30th 2010, 08:19 AM

I don't see how we could just make that step...

$\frac{dy}{d\hat{t}}=-\frac{dy}{dt},\frac{dx}{d\hat{t}}=-\frac{dx}{dt}$

Gives...

$-\frac{dx}{d\hat{t}} = y(t)$
$-\frac{dy}{d\hat{t}}= Ay(t)-x(t)(1-x(t))$

I don't see how we could just subtitute $t= \hat{t}$ in the eqations...

**Ackbeet** · September 30th 2010, 08:22 AM

Do x and y represent physical quantities of any sort?

**Dinkydoe** · September 30th 2010, 08:24 AM

No, they do not specifically represent anything...

So, if we want to rewrite a system in $\hat{t}$ , i don't see why $y(t)$ can simply be replaced by $y(\hat{t})$

**Ackbeet** · September 30th 2010, 08:27 AM

No, they do not specifically represent anything...

A pity, or we could invoke this.

I don't have any other answer, I'm afraid, other than I think you can just do it. I don't have a justification. This is the physicist part of me talking, I freely admit.

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