# Question 6.9 Taylor: Classical Mechanics

1. Jan 3, 2017

### Lujz_br

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

2. Relevant 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 = ex 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 dont get good answer without look at the final of the book.
Thanks! Luiz

2. Jan 3, 2017

### vela

Staff Emeritus
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.

3. Jan 3, 2017

### Lujz_br

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