Results 1 to 2 of 2

Thread: 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

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

    b) Find the particular solution when

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

  2. #2
    Super Member Rebesques's Avatar
    Joined
    Jul 2005
    From
    My house.
    Posts
    658
    Thanks
    42
    Quote Originally Posted by Creebe View Post
    a) Find the general solution of the equation

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

    b) Find the particular solution when

    $\displaystyle u(x,1)=x$.


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

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


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

Search Tags


/mathhelpforum @mathhelpforum