- #1
Rasalhague
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An autonomous ODE is simply an ODE in which the independent variable does not appear explicitly. (hfshaw, Yahoo Answers)
Okay, good, so y' = 3y is an autonomous ODE, while y'(t) = 3y(t) is not autonomous??
In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not depend on the independent variable. (Wikipedia)
Seems like a contradiction in terms. Differentiation of a function with respect to a variable on which it doesn't depend, i.e. one which doesn't denote its argument, is meaningless. I might as well differentiate f:R-->R, f(x) = x2 with respect to the colour of my eyes, instead of with respect to x. So in what sense "does not depend"?
Moreover, any system can always be reduced to a first-order system by changing to the new set of dependent variables y = (x,x(1),x2,...,x(k-1). This yields the new first order system
y' = (y2,y3,...,yk,f(t,y)).
We can even add t to the dependent variables z = (t,y), making the right-hand
side independent of t
z' = (1,z3,z4,...,zk+1,f(z)).
Such a system, where f does not depend on t, is called autonomous. (Teschl, p. 7)
Hmm, let's see. On p. 9, Teschl gives the "simplest (nontrivial) case of a fi rst-order autonomous equation", (1.20)
x' = f(x), x(0) = x0, f in C(R).
Since x is a function in C(J), J subset of R (see p. 6), definition (1.20) is a syntax error. I guess what it means is x' = f o x, where o denotes composition. This looks rather like Wikipedia's definition, an autonomous ODE is an equation of the form
d/dt x(t) = f(x(t)),
which it contrasts with an non-autonomous equation, one of the form
d/dt x(t) = g(x(t),t)).
But equations are normally given not in terms of explicit functions but in a form such as
d/dt x(t) = sin(x(t)).
If I define a particular g by the rule g(p,q) = sin(p), we have
d/dt x(t) = g(x(t),t) = sin(x(t)),
a non-autonomous ODE, according to Wikipedia. Can someone suggest an unambiguous definition of autonomy, or give me any hint? The idea seems to be that the dependence of the equation on t must "pass through" the unknown function; for the equation to qualify as autonomous, Teschl's outer function f must be "blind to t", and only able to perceive it through the medium of x. To get around the problem of whimsical definitions like g(p,q) = sin(p), I tried expressing the idea as "autonomous iff there exists a function such that it doesn't depend on t", but then realized that won't do, given that any equation can be put into autonomous form. Or maybe such a definition is possible, if only I knew which kinds of algebraic rearrangement/rewriting constitute "changing" the form of an ODE, and which are considered trivial.