There are two options, both of which lead to the conclusion that this functional has no stationary points.
One option is to use the natural boundary condition $(L_{y'})'(a)=0$ for each endpoint at which the function is unconstrained. In the present case we have $(L_{y'})'=y''+y'$, so this yields the contradictory conditions $1+C_1=0$ at $x=0$ and $2+C_1=0$ at $x=1$.
Alternatively, you can calculate the value of the functional as a function of the parameters $C_1$ and $C_2$ and minimize it using ordinary calculus. This would be a tedious process if you carried it out in detail, but you can avoid the effort by noting that the value of the functional is linear in $C_2$ with non-zero first-order coefficient, so it doesn't have a minimum with respect to $C_2$.