Orthogonal trajectory problem
I'm supposed to find the orthogonal trajectory of the family of curves of k*x^2+y^2=1. K is constant.The book's anser is ln|y|-(1/2)*y^2 = (1/2) * x^2 + C.
I am getting nowhere near that.
I start by implicit differentiation, 2*k*x + 2*y*y' = 0. Factor out a two, divide both sides by two, and get rid of it. I then have k*x+y*y' = 0. Divide both sides by y, I have (k*x)/y + y' = 0. Subtract the first term from both sides, I then get y'=-(k*x)/y. I take the negative reciprocal, I get y'=y/(k*x).
Switch y' to dy/dx, move all Y to one side and X to the other, and integrate, I end up with ln|y|=k*ln|x|+C, exponentiating I get y=x^k+e^C, which is not the answer in the book.
What am I doing wrong?