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On the exam I was asked the question about Transcritical bifurcation. I gave the equation

$$ \dot x = rx - x^2 $$

Then I was asked why it is not a differential equation and I couldn't answer. I thought if some derivative equals a function - it is a differential equation.

It's not a differential equation, because r is a variable not x?*

*Sorry If something is not clear. I suffer for lack of math in english.

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Why not? If $r$ is a parameter, this is a differential equation. –  Riccardo.Alestra Mar 5 '12 at 8:50
    
Not sure but in bifurcation analysis r is a variable. –  lukas Mar 5 '12 at 8:58
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It is not linear differential equation but it is a differential equation because of x'. –  Mathlover Mar 5 '12 at 9:27
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2 Answers

Definition 1. A differential equation is an equation that involves the derivatives of an unknown function of one or more variables. (Spiegel)

I personally like to change involves by relates.

Since we have an unknown function $x(t)$ and an equation that involves the function $x$ and its derivative:

$$x'(t) = rx(t)-x(t)^2$$

that is a differential equation, and the function you're looking for is

$$x(t) = \frac{r \cdot c \cdot e^{rt}}{1+c\cdot e^{rt}}$$

where $c$ is arbitrary.

I think you might have a bone to pick with your examiner.

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The equation $$\dot{x}(t)=rx(t)-x^2$$ is a family of $ODEs$ because the parameter $r$ can assume infinite values. So it's not an equation, but a family of infinite equations.

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Correction: $r$ cannot assume infinite values, though it can assume any of infinitely many values. (To say that $r$ can assume infinite values is to say that it can take on values like $\infty$.) –  Brian M. Scott Mar 5 '12 at 10:42
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I disagree: $\dot{x}(t)=rx(t)-x^2$ without any further qualification is a differential equation. To specify the family, one must indicate that one is considering the whole family over some range of values of $r$. –  Brian M. Scott Mar 5 '12 at 10:44
    
@Brian M. Scott: I agree. It depends on my bad english. –  Riccardo.Alestra Mar 5 '12 at 11:24
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I figured as much, and I wouldn’t have said anything, except that I’ve heard native speakers make the same mistake (‘infinite values’ instead of ‘infinitely many values’), so I thought that I’d better mention it. –  Brian M. Scott Mar 5 '12 at 11:28
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I'm no expert in differential equations, but to my outsider eye this seems like an unusually pedantic distinction to make. For instance, if I said "the function $f(x) = ax+b$" and someone replied "No, that's not a function. It's a family of functions depending upon the parameters $a$ and $b$," I would probably roll my eyes. Is there something going on here that makes this analogy inappropriate? –  Pete L. Clark Mar 5 '12 at 12:01
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