1
$\begingroup$

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.

  • 2
    It is not linear differential equation but it is a differential equation because of x'.2012-03-05

2 Answers 2

3

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.

0

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.

  • 0
    @PeteL.Clark: it's the only way to defend the statement the equation above is not an ODE2012-03-08