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

  1. #1
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    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.
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  2. #2
    Super Member Rebesques's Avatar
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    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...
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