How to find the function for which ∫ √x*√(1+y'^2) dx is stationary?

[nafizamin]

this is an euler lagrange equation problem from the book- "classical mechanics-John R. Taylor", problem-6.11

find the path function for which ∫ √x*√(1+y'^2) dx is stationary.

the answer is- x= C+(y-D)^2/4C, the equation of a parabola.

here the euler lagrange equation will work on f=√x*√(1+y'^2).

since ∂f/∂y= 0, so ∂f/∂y'= const

→ √x*y'/ √(1+y'^2)= constant.

then i don't get how i get from here to the equation of the parabola.

any help?

 

[voko]

Square both sides of the equation, then do algebra till you get $y'$ on one side, and $x$ on the other. Integrate.

 

[nafizamin]

well, squaring gives me- x*y'^2=C(1+y'^2). i could separate to get x=C(1+y'^2)/y'^2

how do i integrate the terms involving y'^2 ?

 

[voko]

You have not simplified it enough. You can transform that to an equation that does not involve fractions.

 

[srhelfrich]

Does anyone know why the problem asks for y=y(x) but the solution is in the form x=x(y)?