Conflicting results in finding orthogonal curvesI'm told to find the set of orthogonal curves to the curve of equation y=cx2. Using implicit differentiation, dydx=2cx. Then, the desired curves obey the differential equation dydx=−12cx. Integrating, we find the curves y=−12clog⁡{|x|}+K, for some real constant K. The thing is in some solution I found, they rewrite 2cx as 2yx, because c=yx2, and in doing so they get a different differential equation with different curves (y=±K−12x2). Which are clearly different curves. Plotting them on geogebra makes it seem like mine is wrong. What error did I make?

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Conflicting results in finding orthogonal curvesI&#039;m told to find the set of orthogonal curves to the curve of equation y=cx2. Using implicit differentiation, dydx=2cx. Then, the desired curves obey the differential equation dydx=−12cx. Integrating, we find the curves y=−12clog⁡{|x|}+K, for some real constant K. The thing is in some solution I found, they rewrite 2cx as 2yx, because c=yx2, and in doing so they get a different differential equation with different curves (y=±K−12x2). Which are clearly different curves. Plotting them on geogebra makes it seem like mine is wrong. What error did I make?

Ireland Vaughan · Accepted Answer

Step 1Find the set of orthogonal curves to the curve of equation y=cx2.c  would be a fixed constant value, and you looking for a family of curves orthogonal to that single parabola. Finding all such curves is effectively impossible, because away from the parabola there is no requirement at all on the behavior of the orthogonal curves. They can twist and contort in uncountably infinitely many ways.Your solution failed to find all such curves because you applied your differential equationdydx=−12cxnot just at the points (x,cx2) along this fixed parabola, but over the entire plane. As such you got the family of curves that are not just orthogonal to y=cx2, but to every solution of dydx=2cx. That is, you found the family of curves that are orthogonal to the entire family {x↦(x,cx2+c1)∣c1∈BR}.Evidently the problem you are intended to solve isStep 2Find the family of curves orthogonal to y=cx2 for all c∈BR.Your equation dydx=−12cx gives a curve that is orthogonal to one of those curves - the curve for the value of c appearing in that equation. But it does not require the curve to be orthogonal to y=c′x2 for any c′≠c.But when the differential equation is modified to remove c, we get an equationdydx=2yxwhose solutions are exactly the family {x↦(x,cx2)∣c∈BR}. And the orthogonal equationdydx=−x2ygives the set of curves that are orthogonal to all of the family, not just one.

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