Consider $\mathcal{L}=\partial_{xx}+c\partial_x+a$ on $[-L,L]$ with periodic boundary conditions.
Consider the ODE $\lambda u=\mathcal{L}u$.
The characteristic equation then is $$ v^2+cv+a-\lambda=0 $$ which has two solutions, say $v_1(x), v_2(x)$. The general solution therefore is $$ u=c_1e^{v_1 x}+c_2e^{v_2 x}. $$
It is said that from the boundary condition we get $$ c_1+c_2=c_1e^{2Lv_1}+c_2e^{2Lv_2},~~v_1c_1+v_2c_2=v_1c_1e^{2Lv_1}+v_2c_2e^{2Lv_2} $$
Sorry, but I do not see how we get these two equations from our boundary condition.
Could you please explain that to me?