I'm finding this odd since the only functions i've looked at are of the form \(\displaystyle g(x,y,y')\) not \(\displaystyle f(y',y'',y''')\).Write down the Euler-Lagrange equations for critical points of the functional \(\displaystyle I(y)=\int_{x_1}^{x_2}y^2+y'y'''-(y'')^2~dx\)

I only thought the Euler-Lagrange equations were defined the the first type of function, not the second.

Would I do it like this:

\(\displaystyle \frac{\partial f}{\partial y'''}=y'\)

\(\displaystyle \frac{\partial f}{\partial y''}=2(y'')\)

So the E-L equation is \(\displaystyle 2(y'')-\frac{d}{dy'}(y')=0\)

Which becomes \(\displaystyle 2y''-1=0\)