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Math Help - First order ODE form

  1. #1
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    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
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by ScottO View Post
    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: y' = 2x is likely to be of the form y' = f(x)

    while y' = -2xy + 3 is of the form y' = f(x,y)
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  3. #3
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    Quote Originally Posted by Jhevon View Post
    f(x,y) is a multivariable function where we are considering a function of two variables x and y

    y' = -2xy + 3 is of the form 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 y' = -2xy + 3.

    -Scott
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by ScottO View Post
    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 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.
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  5. #5
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    Quote Originally Posted by ScottO View Post
    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 y = f(x), and then of more than one variable 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 t.

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

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

    \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.
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  6. #6
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    Quote Originally Posted by ScottO View Post
    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 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.
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  7. #7
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    Quote Originally Posted by AfterShock View Post
    Hello Scott,

    Well, you know that a function of a single variable is written y = f(x), and then of more than one variable 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 t.
    Ah!. My thinking is just locked into functions like 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 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
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  8. #8
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    OK, to follow up with some related questions...

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

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

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