The function graphed here cannot possibly be a solution to this type of differential equation.
Here's why: This differential equation says that the derivative,
- the rate of change of X, is only a function of X, and that's not the case for this function.
Here's a way to see that: So when X equals 5, here the function is increasing - that's fine,
but here there's a contradiction - the function is decreasing.
If the rate of change depends only on X,
it can't have two different values at the same value X equals 5.
This is an important point, so let me write this out.
Again, the rate of change of X, depends only on X,
- that's what this equation says: the rate of change of X, is only a function of X.
Another way to say that is: the same X value - if I put the same X value in here,
say 5, it has to give the same rate of change,
but, it clearly doesn't here - I put in 5, and I get a positive rate of change,
I put in 5, I get a negative rate of change, here I get another positive one, and so on.
So, this wriggling function cannot possibly be a solution to this equation
and I can state that with certainty, even though I don't know what this f(X) might be,
and this is a general result: differential equations of this form can never have oscillatory solutions,
so, although these are important in engineering and science,
they're a little bit boring from a dynamical systems point of view,
- a particular X curve can only increase, or only decrease, or be at a fixed point. It can never turn around.