Results 1 to 4 of 4

Math Help - Partial Derivatives....wave equation

  1. #1
    Newbie
    Joined
    Sep 2008
    Posts
    14

    Partial Derivatives....wave equation

    Verify that u = t /(a^2 t^2 - x^2) satisfies the wave equation utt = a^2 uxx (Subscript tt and xx)


    I've tried many different ways in deriving this...can't seem to get the answer! I tried the quotient rule, but it didn't work...I just tried the product rule by bringing up the denominator as (a^2 t^2 - x^2)^-1 but I get an x that I cannot get rid of!

    Help would be appreciated!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member flyingsquirrel's Avatar
    Joined
    Apr 2008
    Posts
    802
    Hello,
    Quote Originally Posted by Qt3e_M3 View Post
    Verify that u = t /(a^2 t^2 - x^2) satisfies the wave equation utt = a^2 uxx (Subscript tt and xx)
    Using partial fractions decomposition, u(x,t)=\frac{1}{2a}\left( \frac{1}{at-x}+\frac{1}{at+x}\right). Now differentiate u twice with respect to t, differentiate u twice with respect to x and you're done. If you keep the derivatives under the form \text{constant}\times \left( \text{a fraction}+\text{another fraction}\right) it should be easier than the methods you've tried ; you simply need to know that the derivative of \frac{1}{f} is -\frac{f'}{f^2}.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2008
    Posts
    14
    Quote Originally Posted by flyingsquirrel View Post
    Hello,

    Using partial fractions decomposition, u(x,t)=\frac{1}{2a}\left( \frac{1}{at-x}+\frac{1}{at+x}\right). Now differentiate u twice with respect to t, differentiate u twice with respect to x and you're done. If you keep the derivatives under the form \text{constant}\times \left( \text{a fraction}+\text{another fraction}\right) it should be easier than the methods you've tried ; you simply need to know that the derivative of \frac{1}{f} is -\frac{f'}{f^2}.
    Where did the 1/2a come from?

    I realise that your u(x,t) is equivalent to my u...I'm just not sure how you transformed it into that

    Thanks for the reply!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by Qt3e_M3 View Post
    Where did the 1/2a come from?
    Because \frac{1}{at - x} + \frac{1}{at+x} = \frac{2at}{a^2t^2 - x^2}.
    There is a factor of 2a, you need to cancel it.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: January 27th 2011, 09:08 AM
  2. Partial derivatives, equation help
    Posted in the Calculus Forum
    Replies: 11
    Last Post: December 6th 2010, 02:00 PM
  3. Partial differential equation-wave equation(2)
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: September 6th 2009, 09:54 AM
  4. Partial differential equation-wave equation - dimensional analysis
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: August 28th 2009, 12:39 PM
  5. Second Order Partial Differential Equation (Wave Eqn)
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: March 29th 2009, 10:43 AM

Search Tags


/mathhelpforum @mathhelpforum