Well, it has to be true for every point on the graph of the function. Hence, can't you just use the DE
Find all curves in the first quadrant such that for any point, the segment of the tangent going from the y intercept to the x intercept is bisected by the point of tangency.
Here's my thoughts ...
Let be the point of tangency.
The tangent has the equation
The condition that the segment of the tangent going from the y-intercept to the x-intercept be bisected at the point of tangency means:
... or putting gives the same.
Now I'm stuck.
Thanks for any help!