Need help on Orthogonal Trajectories in my Diff. EQ. Class

[killermike01]

Homework Statement

Show that the families (x+c1)(x²+y²)+x = 0 and (y+c2)(x²+y²)-y = 0

Homework Equations

For the 2 curves to be orthogonal their slopes should be negative recriprocles.

The Attempt at a Solution

I'm pretty sure that for the first set of curves:

y'(x) = - (2c1 x + 3x²+y[x]²+1)/(2(c1+x)y[x])

and for the second set of curves:

y'(x) = (2x (c2 +y)) / (2c2 y[x] + x² + 3y[x]²-1)

which are not negative recriprocles of each other.

I'm thinking i went wrong somewhere along the lines of finding the deravitive. if anyone could please help me out i'd really appreciate it.


I used Mathematica to get those answers:

For the first set i factored out the original problem then took the deravitive of that:

D[c1 x^2 + c1 y[x]^2 + x^3 + x y[x]^2 + x, x]

then i used the solve command to solve that for y'[x]

For the second set i factored out the original problem, then took the deravitive of that with respect to x:

D[c2 x^2 + c2 y[x]^2 + x^2 y[x] + y[x]^3 - y[x], x]

Then i used the Solve[] function to solve that for y'[x]


P.S. I'm pretty sure these are supposed to be orthogonal just because there isn't an option for not orthogonal.

Like i said any help would be appreciated.

 

[Mark44]

I get almost the same as you did, with the difference being a sign in dy/dx for the second curve.

For that curve, I got y' = -(2xy + 2c₂x)/(x² + 3y² + 2c₂y - 1)

I believe that the reason our answers differ is because you weren't working with the equations, which you need to do to differentiate implicitly.

In any case, I didn't find that the derivatives were negative reciprocals of each other, either, so are you sure you copied the problem correctly?