So I can't continue with my work. Don't know if I am just not seeing something under my nose or something.
Given formula: $$I=\int\limits_0^{\pi/2}[(x'_2)^2-(x_2)^2]dt$$
with auxiliary constraints and initial conditions: $$x'_1+x'_2-x'_3=0; x'_1+2x'_3=0$$
$$x_1(0)=0; x_2(0)=0; x_3(0)=0$$ $$x_1(\pi/2)=-\pi; x_2(\pi/2)=3\pi/2; x_3(\pi/2)=\pi/2$$
From here I calculated the Lagrange Equations using the Formula:
$$\frac{d}{dt}(\frac{\partial L}{\partial x'_i})-\frac{\partial L}{\partial x_i} +\sum_{\mu=1}^p (\lambda_\mu [\frac{d}{dt}(\frac{\partial G_\mu}{\partial x'_i})-\frac{\partial G_\mu}{\partial x_i}]+\frac{\partial G_\mu}{\partial x'_i}\frac{d\lambda_\mu}{dt})$$
With $G_1$ and $G_2$ representing the 2 auxiliary constraints and $\lambda$ representing the Lagrange Multipliers, of which there are 2.
The Above Equation was given to me in my textbook. Said Lagrange Equations are: $$\frac{d}{dt}[\lambda_1+\lambda_2]=0$$ $$\frac{d}{dt}[2x'_2-2x_2+\lambda_1]=0$$ $$\frac{d}{dt}[2\lambda_2-\lambda_1]=0$$
From the first and third equation one can see that $\lambda$ is constant because $\lambda'_1=\lambda'_2=0$
From the second equation it stands that $$2x'_2-2x_2+\lambda_1=d$$
where $d$ is a integration constant.
This is where I am stuck.