Results 1 to 3 of 3

Math Help - separating variables! help!

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    7

    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!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,412
    Thanks
    1328
    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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,347
    Thanks
    30
    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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Separating Variables
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 1st 2014, 09:07 PM
  2. [SOLVED] Separating variables - Wave Eqn
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: May 23rd 2011, 11:06 PM
  3. [SOLVED] First Order DE-Separating the Variables
    Posted in the Differential Equations Forum
    Replies: 9
    Last Post: January 18th 2011, 02:02 PM
  4. Separating the variables or some other way?
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: April 25th 2010, 07:35 PM
  5. Not by separating variables
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: November 22nd 2009, 06:39 AM

Search Tags


/mathhelpforum @mathhelpforum