# 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?).

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