Results 1 to 4 of 4

Math Help - 1st Order Semi Linear PDE xu_x+yu_y=u+3

  1. #1
    Senior Member bugatti79's Avatar
    Joined
    Jul 2010
    Posts
    461

    1st Order Semi Linear PDE xu_x+yu_y=u+3

    Folks,

    Could anyone check my attached work. Mathematica seems to output u(x,y)=-3+2x for the particular solution...

    Thanks
    Attached Files Attached Files
    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
    If we parametrize x and y we have

    x(s) \text{ and } x(0)=1

    and

    y(s) \text{ and } y(0)=y_0

    Then

    u(x(s),y(s))

    Now if we take the derivative with respect to s we have

    \frac{d}{ds}u(x(s),y(s))=\frac{dx}{ds}u_x+\frac{dy  }{ds}u_y=u+3

    This gives the system of equations

    \frac{dx}{ds}=x, \quad x(0)=1

    \frac{dy}{ds}=y, \quad y(0)=y_0

    \frac{du}{ds}=u+3

    Solving this gives

    x=e^{s} \quad y=y_0e^{s} \quad u=Ce^{s}-3

    Now if we sub out

    e^{s}

    we get

    u(x,y)=Cx-3 \implies u(1,y)=-1=C-3 \iff C=2

    u(x,y)=2x-3
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,364
    Thanks
    39
    Here's your mistake

    First let e^{f} = g so u = x g\left(\dfrac{y}{x}\right) - 3. If u(1,y) = -1 then -1 =  g(y) - 3 so g(y) = 2 so u = x \cdot 2 - 3 (your Mathematica and TheEmptySet's solution)
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member bugatti79's Avatar
    Joined
    Jul 2010
    Posts
    461
    Cheers Danny!

    I like this alternative method, TheEmptySet. Thanks
    Last edited by bugatti79; May 5th 2011 at 11:52 PM. Reason: clarification
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Form of Semi Linear Linear PDE's
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: April 22nd 2011, 06:51 AM
  2. [SOLVED] Semi Linear PDE
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: February 15th 2011, 02:55 AM
  3. Replies: 11
    Last Post: August 8th 2010, 02:31 AM
  4. Third Order Semi Implicit Runge Kutta Method
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: May 2nd 2009, 07:33 AM
  5. First Order Semi-Linear PDE
    Posted in the Calculus Forum
    Replies: 8
    Last Post: November 15th 2008, 10:26 AM

/mathhelpforum @mathhelpforum