Find the orthogonal trajectory of the system .
Write as , and differentiate with respect to θ: , where . Substitute and simplify a bit, to get .
The orthogonal trajectory will therefore have . Integrate this: .
To find the integral on the right-hand side, substitute , and it becomes .
That gives the orthogonal trajectory as const., or const.
Finally, the trig identity can be used to write the answer as const.