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Hyperbolic Equation, Elliptic Equation
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[QUOTE="psholtz, post: 2895123, member: 93526"] [h2]Homework Statement [/h2] 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] [h2]The Attempt at a Solution[/h2] 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? [/QUOTE]
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