It is well known, that the characterizing property of Clothoids is, that their curvature is proportional to length; that is also the reason, why they are used as design elements e.g. in road design.

Question:
Has the analogue to clothoids, where the slope and not the curvature is proportional to length, ever been described or investigated?

The associated differential equation $$y'(x) =\int_0^x \sqrt{1+y'(t)} dt $$ isn't hard to devise and the curve might also have practical applications also e.g. in road design.

I am specifically looking for articles on the solution of the associated differential equation, but other articles related to the curve are also welcome.

It is well known, that the characterizing property of Clothoids is, that their curvature is proportional to length; that is also the reason, why they are used as design elements e.g. in road design.

Question:
Has the analogue to clothoids, where the slope and not the curvature is proportional to length, ever been described or investigated?

The associated differential equation $$y'(x) =\int_0^x \sqrt{1+y'(t)^2} dt $$ isn't hard to devise and the curve might also have practical applications also e.g. in road design.

I am specifically looking for articles on the solution of the associated differential equation, but other articles related to the curve are also welcome.