Results 1 to 2 of 2

Math Help - Partial derivatives (from D'Alembert)

  1. #1
    Newbie
    Joined
    Nov 2010
    Posts
    8

    Partial derivatives (from D'Alembert)

    Hello, I'm wondering, how would you answer the following question?



    I've put in the conditions, but really can't see how to find the functions. I'd really appreciate some help. Thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by jonmondalson View Post
    Hello, I'm wondering, how would you answer the following question?



    I've put in the conditions, but really can't see how to find the functions. I'd really appreciate some help. Thank you.
    So we have that

    u_t(x,t)=-cF'(x-ct)+cG'(x+ct)

    If you plug into the equations you get that

    \frac{1}{x^2+1}=F(x)+G(x)

    0=-cF'(x)+cG'(x) \iff F'(x)=G'(x)

    The last line gives

    F(x)=G(x)+k

    This gives

    G(x)=\frac{1}{2}\frac{1}{x^2+1}-\frac{k}{2}

    and

    F(x)=\frac{1}{2}\frac{1}{x^2+1}+\frac{k}{2}

    F(x-ct)=\frac{1}{2}\frac{1}{(x-ct)^2+1}+\frac{k}{2}

    G(x+ct)=\frac{1}{2}\frac{1}{(x+ct)^2+1}-\frac{k}{2}

    So the final solution is

    u(x,t)=\frac{1}{2}\frac{1}{(x-ct)^2+1}+\frac{1}{2}\frac{1}{(x+ct)^2+1}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Partial Derivatives
    Posted in the Calculus Forum
    Replies: 3
    Last Post: June 19th 2011, 02:46 PM
  2. Partial Differential Equation using D'Alembert's approach
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: February 23rd 2009, 02:55 PM
  3. partial derivatives
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 7th 2008, 10:08 AM
  4. partial derivatives
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 7th 2008, 06:58 AM
  5. Partial derivatives
    Posted in the Calculus Forum
    Replies: 7
    Last Post: March 6th 2007, 07:22 PM

Search Tags


/mathhelpforum @mathhelpforum