Question

Differential equations to find specific curves

Answer 1

Neat little problem, I've not seen that one before - cheers :).

Draw a simple diagram of it, recall the geometric definition of a gradient, construct two right-angled triangles with hypotenuses being the two distances of interest, take the squares of them, and I think you want to solve:

f(x)^2 + (f(x)/f'(x))^2 = x^2 + f(x)^2 .  

Don't forget the - as well as the +

Answer 2

Remember sqrt(x^(2)) = |x|. So at some point in the calculation, you get multiple cases. (Technically 4 cases, but some equations end up equivalent, so 2.) Breaks down to y'=+/- y/x