So I decided to use the Euler Lagrange equations:Find solutions of the Euler-Lagrange equations for critical points of the following functionals with the given endpoint conditions.

$\displaystyle I(y)=\int_1^2 x(y')^2~dx$ where $\displaystyle y(1)=1$ and $\displaystyle y(2)=2$

$\displaystyle \frac{\partial f}{\partial y}=0$ and $\displaystyle \frac{\partial f}{\partial y'}=2(y')x$

Inputting this into the Euler-Lagrange equations gives:

$\displaystyle -\frac{d}{dx}(2y'x)=0 \ \Rightarrow -2y'=0 \Rightarrow y'=0$

So integrating both sides w.r.t x gives $\displaystyle y=A$ where $\displaystyle A$ is a positive constant.

From here I was expecting to use the conditions on y to work out what A is. Unfortunately this doesn't work! y can't equal a constant and yet change from 1 to 2 as described in the conditions.

What's going wrong?