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? 
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> 
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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. 



"Slope Analogue" of Clothoids
Note that $y''(x)=\sqrt{1+y'(x)^2}$ so $u=y'$ satisfies $u'(x)=\sqrt{1+u^2}$. This is separable, so we get
$$
\frac{du}{\sqrt{1+u^2}}=dx,\quad \sinh^{-1}(u)=x + C.
$$


