# How to solve this differential equation involving an integral

I am working on a problem involving finding a function for which the area under the curve is equal to the arc length. I therefore come up with $$\int_a^b(k\sqrt{1+[f'(x)]²} -f(x))dx =0$$. Frankly, I have no idea where to even start to solve this. I have read that it may involve hyperbolic functions, but I'm not really sure.

• This looks more like a calculus of variations problem than differential equations. – Paul Aug 14 at 13:44
• Is this an identity in $a$ and $b$ ? Also, what can you tell us about $k$, from the description it looks like $k=1$. – WW1 Aug 14 at 14:26
• k is a real constant, and so are a and b. – John Arg Aug 16 at 12:04