# Thread: First order ODE form

1. ## First order ODE form

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

2. Originally Posted by ScottO
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
f(x,y) is a multivariable function where we are considering a function of two variables x and y

f(x) means we are considering a single variable function of x

example: $\displaystyle y' = 2x$ is likely to be of the form $\displaystyle y' = f(x)$

while $\displaystyle y' = -2xy + 3$ is of the form $\displaystyle y' = f(x,y)$

3. Originally Posted by Jhevon
f(x,y) is a multivariable function where we are considering a function of two variables x and y

$\displaystyle y' = -2xy + 3$ is of the form $\displaystyle y' = f(x,y)$
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 $\displaystyle y' = -2xy + 3$.

-Scott

4. Originally Posted by ScottO
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 $\displaystyle y' = -2xy + 3$.

-Scott
oh no, PDE's are a different creature from what i was mentioning here. i don't know much about PDE's.

5. Originally Posted by ScottO
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
Hello Scott,

Well, you know that a function of a single variable is written $\displaystyle y = f(x)$, and then of more than one variable $\displaystyle y = f(x,y)$, 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 $\displaystyle t$.

In classifying a Dif EQ, you have the type, order and linearity. Both the following are in "normal form"

$\displaystyle \frac{dy}{dt} = f(t,y)$

$\displaystyle \frac{d^2y}{dt^2} = f(t,y,y')$

They represent the first and second order ODEs.

Let me know if that makes things a little clearer.

6. Originally Posted by ScottO
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 $\displaystyle y' = -2xy + 3$.

-Scott
Generally, in a first year Dif EQ course, only ODE's are considered. They are functions of 1 variable. Then, PDE's or partial Dif EQ's have more than one variable.

7. Originally Posted by AfterShock
Hello Scott,

Well, you know that a function of a single variable is written $\displaystyle y = f(x)$, and then of more than one variable $\displaystyle y = f(x,y)$, 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 $\displaystyle t$.
Ah!. My thinking is just locked into functions like $\displaystyle f(x) = y = mx + b$. Makes sense to me now, since the whole idea of differentials is to express rates of change. This explains why sometimes I see $\displaystyle t$'s pop up, "out of nowhere", after an integration or some other step in the discussion.

Let me know if that makes things a little clearer.
Yes, thanks. And to you as well, Jhevon.

-Scott

8. OK, to follow up with some related questions...

A very general ODE form: $\displaystyle F(x, y, y', y'', ..., y^n) = 0$

My simpler case: $\displaystyle y' = f(x,y)$

In either of these forms, is it correct to view each of the terms, $\displaystyle x, y, y'$, etc. as whole functions? For example, the $\displaystyle x$ could be just $\displaystyle x$, or something more like $\displaystyle x^2+3x$. 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