Extraordinary Differential Equations #8

Calculus Level pending

Orthogonal trajectories are a family of curves (say \(F_2\)) in a plane that intersect a given family of curves (say \(F_1\)) at right angles.

Suppose \(F_1\) is given by \[y=c e^{-x^2}\] where \(y=y(x)\) and \(c\) is an arbitrary constant. Derive an equation for \(F_2\) where \(c\) is an arbitrary constant of integration.

Hint: If \(F_1\) is defined by a differential equation of the form \(y'=f(x,y)\), then \(F_2\) is defined by the differential equation \(y'=-\frac{1}{f(x,y)}\) (why?).

(This problem is part of the set Extraordinary Differential Equations.)

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