- #1

psholtz

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## Homework Statement

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]

## 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?