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Math Help - separating variables! help!

  1. #1
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    Lightbulb separating variables! help!

    Sorry, but this is an upcoming exam question for me. I need help!(and I only got a short time to find out)

    Find solutions of the following equation by separating variables:

    <br />
\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0


    Thanks to whoever that can help!
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  2. #2
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    Quote Originally Posted by ntfabolous View Post
    Sorry, but this is an upcoming exam question for me. I need help!(and I only got a short time to find out)

    Find solutions of the following equation by separating variables:

    <br />
\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0


    Thanks to whoever that can help!
    Do you know what "separating variables" means? If you have a test coming up, this is hardly the time to start learning the subject!

    It means assuming a solution of the form u(x,y)= A(x)B(y) where A and B are functions of x alone and y alone, respectively. Then \frac{\partial u}{\partial x}= \frac{dA}{dx}B and \frac{\partial u}{\partial y}= A\frac{dB}{dy}.

    So that equation becomes \frac{dA}{dx}B+ A\frac{dB}{dy}= 0. That can be written as \frac{dA}{dx}B= -A\frac{dB}{dy} and, dividing both sides by AB, \frac{1}{A}\frac{dA}{dx}= -\frac{1}{B}\frac{dB}{dy}.

    Now, the left side is a function of x only and the right side is a function of y only (we have "separated" the variables) so the only way that can be equal for all x and y is if they are both equal to the same constant.

    That is, we can separate into two equations: \frac{1}{A}\frac{dA}{dx}= \lambda or \frac{dA}{dx}= \lambda A and -\frac{1}{B}\frac{dB}{dy}= \lambda or \frac{dB}{dy}= -\lambda B.

    There is no way to determine what " \lambda" is from the equation alone. Depending upon the additional (boundary or intial value) conditions, the solution might be the product of A and B for a specific \lambda or a sum of such products for many (possibly infinitely many) values of \lambda.
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  3. #3
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    Quote Originally Posted by ntfabolous View Post
    Sorry, but this is an upcoming exam question for me. I need help!(and I only got a short time to find out)

    Find solutions of the following equation by separating variables:

    <br />
\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0


    Thanks to whoever that can help!
    I might also add the one could seek solutions of the form

    u = A(x) + B(y)

    which is also a separation of variables. BTW - the general solution of the above is u = f(x-y) for any f.
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