Applications of the Euler-Lagrange Equation

aaj92

1. The problem statement, all variables and given/known data

Find and describe the path y = y(x) for which the integral [itex]\int[/itex][itex]\sqrt{x}[/itex][itex]\sqrt{1+y^{' 2}}[/itex]dx (the integral goes from x1 to x2.... wasn't sure how to put that in. sorry) is stationary.

2. Relevant equations

[itex]\frac{\partial f}{\partial y}[/itex] - [itex]\frac{d}{dx}[/itex][itex]\frac{\partial f}{\partial y^{'}}[/itex]=0

3. The attempt at a solution

[itex]\frac{\partial f}{\partial y}[/itex] = 0 so that means [itex]\frac{d}{dx}[/itex][itex]\frac{\partial f}{\partial y^{'}}[/itex] also has to equal zero. this means that [itex]\frac{\partial f}{\partial y^{'}}[/itex] is a constant. So I set it equal to a random constant c and solved for x. I ended up getting x = 2C[itex]^{2}[/itex] which is really wrong :p. The answer is supposed to be x = C + [itex]\frac{(y-D)^{2}}{4C}[/itex]
where C and D are both constants... I don't understand how they ended up with two constants. I'm not really sure how the constants work. In class it seems like he just sort of pulls them out of the air and so now I'm confused