Results 1 to 2 of 2

Math Help - First Order, Quasi-Linear, Partial Differential Equations

  1. #1
    Junior Member
    Joined
    Mar 2009
    Posts
    31

    First Order, Quasi-Linear, Partial Differential Equations

    a) Find the general solution of the equation

    (x^2 - y^2 - u^2)u_x + 2xyu_y = 2xu

    b) Find the particular solution when

    u(x,1)=x.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Rebesques's Avatar
    Joined
    Jul 2005
    From
    At my house.
    Posts
    538
    Thanks
    11
    Quote Originally Posted by Creebe View Post
    a) Find the general solution of the equation

    (x^2 - y^2 - u^2)u_x + 2xyu_y = 2xu

    b) Find the particular solution when

    u(x,1)=x.


    The characteristic system is \begin{cases} dx/(x^2-y^2-u^2)=dy/2xy  \  (1)\cr du/dy=u/y \ \  \ (2)\end{cases}.

    Equation (2) gives u=cy, so substituting into (1) we obtain
    dx/dy=(x/2y)-[(1+c^2)y/2x]. Solve this (how?) to get
    x^2=2y^2(2c^*y-d), where c^*=c^*(c)=-(1+c^2)/2. It all eventually leads to x^2/2=-(1/2)-(u^2y/2y^2x)+d,
    and use the initial conditions \{y=1,u=x\} to determine d=d(x).


    Oh and check the calculations...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Linear first order differential equations
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: February 18th 2011, 04:40 AM
  2. First Order Linear Differential Equations Problems
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: September 3rd 2010, 12:01 AM
  3. First order differential and linear equations
    Posted in the Calculus Forum
    Replies: 4
    Last Post: June 16th 2009, 03:56 PM
  4. Second Order Linear Differential Equations
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: February 15th 2009, 10:15 AM
  5. Replies: 27
    Last Post: February 10th 2008, 05:49 PM

Search Tags


/mathhelpforum @mathhelpforum