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Homework Help: Hyperbolic Equation, Elliptic Equation

  1. Sep 22, 2010 #1
    1. The problem statement, all variables and given/known data
    From the relation:

    [tex]A(x^2+y^2) -2Bxy + C =0[/tex]

    derive the differential equation:

    [tex]\frac{dx}{\sqrt{x^2-c^2}} + \frac{dy}{\sqrt{y^2-c^2}} = 0[/tex]

    where [tex]c^2 = AC(B^2-A^2)[/tex]

    3. The attempt at a solution

    I'm able to (more or less) do the derivation, but I think the correct relation for the constants is:

    [tex]c^2 = AC/(B^2-A^2)[/tex]

    My reasoning is as follows.. start with the first relation:

    [tex]u(x,y) = A(x^2+y^2) - 2Bxy + C =0[/tex]

    and differentiate:

    [tex]u_x = 2Ax - 2By = 2(Ax-By)[/tex]

    [tex]u_y = 2Ay - 2Bx = 2(Ay-Bx) [/tex]

    [tex]du = \frac{\partial u}{\partial x}dx + \frac{\partial u}{\partial y}dy =
    (Ax-By)dx + (Ay-Bx)dy =
    0[/tex]

    Solving the quadratic expression u(x,y) for x and y respectively, we can write this equation as:

    [tex] \sqrt{(B^2-A^2)y^2 - AC}dx + \sqrt{(B^2- A^2)x^2 - AC}dy=0[/tex]

    [tex]\frac{dx}{\sqrt{(B^2-A^2)x^2-AC}} + \frac{dy}{\sqrt{(B^2-A^2)y^2-AC}} = 0[/tex]

    [tex]\frac{dy}{(B^2-A^2)^{1/2}\sqrt{x^2-\frac{AC}{B^2-A^2}}} + \frac{dy}{(B^2-A^2)^{1/2}\sqrt{y^2-\frac{AC}{B^2-A^2}}} = 0[/tex]

    [tex]\frac{dx}{\sqrt{x^2-\frac{AC}{B^2-A^2}}} + \frac{dy}{\sqrt{y^2-\frac{AC}{B^2-A^2}}} = 0[/tex]

    and so setting:

    [tex]c^2 = \frac{AC}{B^2-A^2}[/tex]

    we can write:

    [tex]\frac{dx}{\sqrt{x^2-c^2}} + \frac{dy}{\sqrt{y^2-c^2}} = 0[/tex]

    My question is: was there a typo in the initial problem statement (i.e., the expression for c^2), or did I make a mistake in my derivation?
     
  2. jcsd
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