How are they differentiating this ODE?

[JG89]

They give a differential equation: $x' = f_a(x) = ax(1-x)$. In determining if the equilibrium points are sources or sinks, they say: We may also determine this information analytically. We have $f'_a(x) = a - 2ax$

How can they differentiate with respect to x? x is a function, it doesn't represent a point on the real line. I tried assuming that they really mean $x'(t) = f_a(x(t)) = ax(t)(1 - x(t))$, but that would mean that $x''(t) = f'_a(x(t)) = ax'(t) - 2ax(t)x'(t)$, which according to the book is wrong.

What's going on here?

 

[Simon Bridge]

Differentiating with respect to a function is not a problem.

1st pass - mistake in notation?
Well x'=1 if we take the primed notation to indicate differentiation in x. Perhaps that's supposed to be a dot? Then I can put v = x' = dx/dt right?

That would mean that v = ax(1 - x) and and you can certainly differentiate speed with respect to space to give: dv/dx = a - 2ax

Of course this means that the notation is inconsistent.
I think it's pretty clear that they are differentiating f with respect to x. For the first it's not so clear from the example what is intended ... I mean where they came from is something like x=(0.5)ax²(1-x)-ax+c ... which is only true for at most three values of x. So there is something missing from the description here.

2nd pass: your analysis holds but recalling that x'=dx(t)/dx=1 then x''=0 and your equation simplifies to:

0 = f'(x(t)) = a - 2ax(t)

... isn't that what they have?

3rd pass ... If the prime implies d/dt (JIC) always then ...

f = ax - ax² = x'
f' = ax' -2ax.x' = af -2axf = ax(1-x-2x²) (check - not what they have)

I think we need context but it really looks like they have differentiated f wrt x not t.

 

[JG89]

I know they're differentiating with respect to x, that's exactly what I have a problem with. As far as I know, the usual derivative of a map requires the domain to be a subset of $\mathbb{R}^n$, but the x that they are differentiating with respect to is a function, it's not a real number, or an n-tuple of real numbers. Shouldn't $x in \mathbb{R}^n$ if we're differentiating with respect to x?

 

[Simon Bridge]

So you are telling me that x is not in $\mathbb{R}^n$?
What is it's domain then? You could, in principle, plot a graph of x vs t right?

Are you saying that if y=f(x(t)) you can't do dy/dx?

 

[JG89]

Wow, now I feel silly. I confused the hell outta myself, I should've known better. Thanks for the replies.

 

[Simon Bridge]

No worries - you've been thinking too hard go have a drink.