1. ## Orthogonal trajectory problem

Hi,

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?

1. You should get $\ln|y|=\dfrac{1}{k}\,\ln|x|+C=\ln|x^{1/k}|+C.$

2. When you exponentiate, the additive constant of integration becomes a multiplicative one thus: $y=Cx^{1/k}.$ You can check to see that this satisfies the orthogonal trajectory DE (which I do believe is correct).

The book's answer mystifies me. I think it's the answer for a different problem, honestly. Double-check the problem number.

3. There have been 5 other problems where the book has been completely wrong, and this is apparently one of them. This is the first semester using this book at the college and the instructor had not vetted these problems, but yes, that is the answer the book gives. ln|y-1/2y^2...etc You'd think with this being edition number 12 their proofreaders would have gotten them by now, but anyway...

Thank you for the help!

4. You're welcome!