Grant Erwin wrote:

It's a little more complicated than your analysis, but here 'tis:

In my solution, the final equation looks like

y = Px + Q + SQRT(R*X^2 + S*X + T)

and for a curve going through (0,0) (2,0.75) and (2.5,5) and tangent to the x
axis at (0,0) and very nearly vertical at (2.5,5) I got my 5 constants as:

P = -12.083
Q = 35.208
R = 142.007 ;; there may be slight roundoff or truncation errors in these
S = -850.868
T = 1239.627

It took me an amazing amount of time flailing at this problem (which I didn't at
all need to solve; I just like doing math sometimes) before I resorted to
looking up how aircraft designers used to lay out airframes, and found that back
in the 1950s those guys really knew about 2nd order equations and how to whip
them into shape.

GWE

I get

48*x^2 + 12*y^2 + 290*x*y -845y =0

or

y=[(845-290x)-SQRT(81796*x^2-490100*x+714025)]/24

as a solution where the tangent is vertical at (2.5,5)