Suppose that the area bounded by a smooth curve y(x), the x-axis, the y-axis, and the vertical line through x equals twice the curve's arc length between 0 and x. Find the differential equation satisfieded by the curve.
(Hint: review the formula for arc length from calculus.)

So I know what the arc length formula is but
I'm not sure how to do this problem, any hints or starters will be appreciated! Thanks guys!

The area is : integral between 0 and x of y(t)

The arc length is : integral between 0 and x of sqrt(1+y'(t)²)

A differential equation satisfied by y is :

integral ( 0 ; x ; y(t)dt ) = 2 * integral ( 0 ; x ; sqrt( 1 + y'(t)²dt )

Those 2 functions must have the same derivative functions :

y(t) = 2 * sqrt ( 1 + y'(t)² )

Are you also asked to solve it ?