# Thread: Rewriting a ODE in -t

1. ## Rewriting a ODE in -t

It feels like a stupid question...

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

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

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

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

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

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

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

How do we deal with that?

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

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

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

Wouldn't that do the job?

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

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

Gives...

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

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

6. Do x and y represent physical quantities of any sort?

7. No, they do not specifically represent anything...

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

8. 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.