I have a line that has initially length one. A process reduces the length of the line.
During each step of the process 2 points $x_1$ and $x_2$ on the remaining line are selected. The part of the line between the left side and the maximum of $x_1$ and $x_2$ is cut off.
What is the expected number of times $E(x)$ the process should be repeated such that a fraction x of the initial line has disappeared.
I would like to solve this problem using a differential equation.
I did allready some work on this problem, but I am not sure of the correct DE to solve the problem. More precisely I would like to know if the correct DE to solve this problem is
$E(x) = 1-x^2 + \int_0^x 2y+2yE(\frac{x-y}{1-y}) dy$
or
$E(x) = 1-x^2 + \int_0^x 1+2yE(\frac{x-y}{1-y}) dy$
As you see my problem is that I don't know whether the first term of the integrand should be 1 or 2y.
Some additional info to make it easier to understand the DE's:
$1-x^2$ = probability 1 step is enough to make disappear more than x
$E(\frac{x-y}{1-y})$ = expected number of times the process should be repeated such that a fraction x has disappeared, starting from length y < x
$2y$ = probability distribution function for the maximum of x_1 and x_2 with 0<=y<=1