(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the orthogonal trajectories of the given family of curves:

All circles through the points (1,1) and (-1,-1)

I have reduced the problem to finding solutions to the following differential equation:

[tex]y'=\frac{y^2-2xy-x^2+2}{y^2+2xy-x^2-2}[/tex]

2. Relevant equations

3. The attempt at a solution

I believe the equation for all circles which pass through the given points is:

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

Solving for C:

[tex]C = \frac{-x^2-y^2+2}{2 (x-y)}[/tex]

Implicit Differentiation of the original equation:

[tex]2(x+C) + 2(y-C)y'=0[/tex]

[tex]y'=\frac{x+C}{y-C}[/tex]

So, substituting C and taking the negative reciprocal, orthogonal trajectories must satisfy:

[tex]y'=\frac{y^2-2xy-x^2+2}{y^2+2xy-x^2-2}[/tex]

And now I'm stuck. The answer in the book agrees with the steps I've taken so far, but I have no idea how to get there from here aside from guessing the answer out of thin air.

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# Homework Help: Differential Equation related to orthogonal trajectories

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