Find $\displaystyle y_1(x), y_2(x)$ so that $\displaystyle J$ is minimized.

$\displaystyle J(y_1,y_2)=\int_0^\frac{\pi}{4}(4y_1^2+y_2^2+y_1'y _2')dx$

$\displaystyle y_1(0)=1, y_2(0)=0$

$\displaystyle y_1(\frac{\pi}{4})=0, y_2(\frac{\pi}{4})=1 $

My work so far

I set up the Euler-Lagrange equation for each of the functions, giving me a set of second order differential equations:

$\displaystyle y_1''=2y_2$

$\displaystyle y_2''=8y_1$

I tried putting the equations in matrix form and using Wolfram to calculate the eigenvalues and eigenvectors. But solving a 4x4 system with complex eigenvalues seems to be beyond the scope of my calculus of variations course. I suspect there is a simpler way to do it, something that can be done by hand.

According to my textbook, a first integral of a Lagrangian of several functions, not explicitly dependent on $\displaystyle x$, is given by:

$\displaystyle L-\sum_{i=0}^{n}y_i'L_{y_{i}^{'}} = C$

Using that property I obtained another equation:

$\displaystyle 4y_1^2+y_2^2-y_{1}^{'}y_{2}^{'}=C$

I'm not sure what to do with this though. So basically I'm stuck.

Any help is appreciated!