Thanks Charlie, I can now see my error – my assumption that all solutions must
lie on y = x is wrong. In fact, the curves y = bx and x = by can also intersect at
points off that line and, because of the symmetry about y = x, there would then
be an odd number of solutions, with one on the line and others in symmetric
pairs about y = x.
This is what is happening when 0 < b < e-e, as Broll discovered.
At (e-e, e-e) the curve y = bx crosses y = x with gradient -1 and for b < e-e
there will be three solutions. For example:
when b = 0.05,x ~= 0.35022(y ~= 0.35022) x ~= 0.13736(y ~= 0.66266) x ~= 0.66266(y ~= 0.13736)
So:when 0 < b < e-ethere are 3 solutions whene-e <= b <=1there is 1 solution when1 < b < e1/ethere are 2 solutions when b = e1/ethere is 1 solution whene1/e < bthere are no solutions
Solution - correction
