Question 6.9 Taylor: Classical Mechanics

[Lujz_br]

Homework Statement

Hello, I solved others but not 6.9:
Find the equation of the path joining the origin O to point P(1,1) in the xy plane that makes the integral ∫(y'² +yy' + y²) dx stationary.
∫ from O to P. y' = dy/dx

Homework Equations

I need use ∂f/∂y = d/dx (∂f/∂y') (euler-lagrange equation) with f = y'² +yy' + y²
∂f/∂y = y' + 2y
∂f/∂y' = 2y' + y

and d/dx (∂f/∂y') = 2 y'' + y', go to euler-lagrange equation we get:
y' + 2y = 2 y'' + y'
this is equivalent to:
y'' = y (eq 1)

y = e^x is solution of eq. 1, but it don't fit (0,0) and (1,1)
I see the solution at final of the book: y = senh(x)/senh(1)
Ok, is solution of eq. 1 and fit points (0,0) and (1,1).

There are any thing more? I feel I don't get good answer without look at the final of the book.
Thanks! Luiz

 

[vela]

You have a second-order differential equation, so you should have two solutions. You found one. What's the other one? The general solution will be a linear combination of the two solutions.

 

[Lujz_br]

Ok, 1st is y = A1 e^x 2nd is y = A2 e^-x and general solution is y = A1 e^x + A2 e^-x which go to y = senh(x)/senh(1).
Ok, remember this (two solutions) is fine way to get the right answer... :)