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First Order Nonlinear PDE

  1. Jan 7, 2010 #1
    i have to solve this equation :

    du/dx * du/dy = x*y

    u(x,y) = x for y =0

    with putting this equation in the form : F(x,y,u,du/dx,du/dy) = 0 . it can be solved.
    But mine book does not explain how to do this, there are no examples.

    Can someone help me ? or any links of examples on the internet?
  2. jcsd
  3. Jan 7, 2010 #2
    Your PDE already is in a such form!?

    If you suppose that


    then substitution such u into PDE gives you after separation of variables

    diff(A(x),x)*A(x)/x=1/(B(y)*diff(B(y),y)/y)=c ,

    where c is a constant. Solutions of these two ODEs lead to particular solution

    u(x,y) = (c*x^2+C2)^(1/2)*1/c*(c*(y^2+C1*c))^(1/2)

    (among other things this solution allows to obtain the general sulution to your PDE!).
    If now substitute y=0 into above solution, we find that it must be

    C2=0, C1=1/c

    so the required solution is as follows

    u(x,y) = x*(y^2+1)^(1/2)
  4. Jan 8, 2010 #3
    Thank you for giving the answer. Mine mistake I had to tell, to solve this problem we had to use method of characteristics. I did it and found the same answer. so it is correct.
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