Finding the curve to minimize a functional

[Esran]

Homework Statement

Find the curve y(x) that passes through the endpoints (0,0) and (1,1) and minimizes the functional I[y] = integral(y'² - y²,x,0,1).

Homework Equations

Principally Euler's equation.

The Attempt at a Solution

We choose f{y,y';x} = y'² - y². Our partial derivatives are:

df/dy = -2y
df/dy' = 2y'

Euler's equation gives:

df/dy - d/dx(df/dy') = 0
-2y - 2y'' = 0

The general solution for this differential equation is:

y = A cos(x) + B sin(x)

To find A and B, we use our constraint that y(0) = 0 and y(1) = 1. Our curve is then y(x) = sin(x)/sin(1).

Have I done this problem correctly? If not, where did I go wrong?

 

[diazona]

I don't see anything wrong with your method.