Find the EL eqns of $\displaystyle \int_0^1 (\dot x_1^2 + \dot x_2^2 - k^2(x_1 + x_2)^2)\: dt$ where $x_1 = x_1(t), x_2=x_2(t)$, $k$ constant and solve the system of equations
E-L eqns from: $\displaystyle 0 = \frac{\partial f}{\partial x_i} - \frac{d}{dt}\left(\frac{\partial f}{\partial \dot{x_i}}\right)$ and $f(t,x_1,x_2, \dot x_1, \dot x_2) = \dot x_1^2 + \dot x_2^2 - k^2(x_1 + x_2)^2$
I have \begin{align*} 0 &= \frac{\partial f}{\partial x_1} - \frac{d}{dt}\left(\frac{\partial f}{\partial\dot x_1 }\right)\\ &= -2k^2(x_1 + x_2) - \frac{\partial}{\partial t}2\dot x_1\\ \Longrightarrow 0 &= \ddot x_1 + k^2(x_1 + x_2)\\ \text{and } 0 &= \frac{\partial f}{\partial x_2} - \frac{d}{dt}\left(\frac{\partial f}{\partial \dot x_2}\right)\\ &= -2k^2(x_1 + x_2) - \frac{d}{dt}2\dot x_2\\ \Longrightarrow 0 &= \ddot x_2 + k^2(x_1 + x_2) \end{align*} Is
\begin{align*} 0 &= \ddot x_1 + k^2(x_1 + x_2)\\ 0 &= \ddot x_2 + k^2(x_1 + x_2) \end{align*}
the system of equations?
And how do I find the general solution to this system?
The system of equations you have looks correct to me. To solve the system, note that by setting: $$u=x_1+x_2,\quad v=x_1-x_2$$ We obtain the system (to obtain the first equation, add the two equations in your system, then subtract them for the second equation): $$\ddot{u}+2k^2 u=0$$ $$\ddot{v}=0$$ This is an uncoupled set of ODE's. After solving this, we can recover $x_1(t)$ and $x_2(t)$ using that: $$x_1=\frac{u+v}{2},\quad x_2=\frac{u-v}{2}$$