Hi Folks,
OK, I'm wanting to get a better grip on jargon regarding ODE's.
A very basic form is:
y' = f(x, y).
The books then mention that the simplest case of this form is:
y' = f(x).
So, first question, what exactly is the difference between f(x, y) and f(x)?
-Scott
OK. If I understand correctly, a/the difference between ODE's and partial DE's, is that PDE's have more than one independent variable.
So what is the relationship between being a multi-variable function, and the number of independent variables? Is it determined by the specific problem, so that it could be either? And if so, I guess the equivalent PDE would be written differently than .
-Scott
Hello Scott,
Well, you know that a function of a single variable is written , and then of more than one variable , etc. In Dif EQ, since so many applications in the study of Dif Equations involve time, it's convenient to use t has an independent variable. Then, x and y denote dependent variables - they are understood to be real-valued functions of the independent variable .
In classifying a Dif EQ, you have the type, order and linearity. Both the following are in "normal form"
They represent the first and second order ODEs.
Let me know if that makes things a little clearer.
Ah!. My thinking is just locked into functions like . Makes sense to me now, since the whole idea of differentials is to express rates of change. This explains why sometimes I see 's pop up, "out of nowhere", after an integration or some other step in the discussion.
Yes, thanks. And to you as well, Jhevon.Let me know if that makes things a little clearer.
-Scott
OK, to follow up with some related questions...
A very general ODE form:
My simpler case:
In either of these forms, is it correct to view each of the terms, , etc. as whole functions? For example, the could be just , or something more like . And likewise the other terms.
In other words, the first general form above is really a bunch of functions added together to represent a single, complex DE.
Sorry if these seem like odd question, but I'm studying math on my own as a hobby. So I don't have a professor to decipher the math speak. And text books usually assume you know what all the symbols mean in various contexts.
-Scott