Hi,

I just started learning calculus of variations, and got this problem on the first day:

Given

$\displaystyle F(y)=\int_0^1(1+x)y'^2dx$

and

$\displaystyle y(x)=x+c_1x(1-x)+c_2x^2(1-x)$,

find the $\displaystyle c_1$ and$\displaystyle c_2$ that minimizes$\displaystyle F$.

My approach so far:

First find$\displaystyle y^'$ and $\displaystyle y'^2$, then integrate$\displaystyle (1+x)y'^2$.

Define the answer as a new function $\displaystyle I=f(c_1,c_2)$ and find the point where its partial derivatives are both zero:

$\displaystyle \frac{\partial I}{\partial c_1}=\frac{\partial I}{\partial c_2}=0$

This turns out to be a lot of work, and it's hard not to make any mistakes along the way. So I was wondering if there's a better way to get the solution. A shortcut, if you will.

Thanks,

TwoPlusTwo