Question

Anyone mind taking a look at this and checking my work please?

Answer 1

Check my work: Find the orthogonal trajectories of the given family of curves:

y^2-x^2=c

So I use implicit differentiation to get
dydx=xy


Then, I take the negative reciprocal
dydx=−yx


Next, I separate the terms and integrate so
∫−1ydy=∫1xdx
giving
−lny=lnx+c

Answer 2

exponentiating both sides, Ive got:
1/y=xec
. Now I just call e^c C, because its still some constant. So now I've got
1/y=cx
and then just divide by x to get c alone. So i end with
1/xy=c
How does this look? Am I close?