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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?

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  • $\begingroup$ Perhaps the interval is $[0,2L]$? The boundary condition is obtained by $u(0)=u(2L)$ and $u'(0)=u'(2L)$, where $u'$ is the derivative of $u$ w.r.t. $x$. $\endgroup$ – Aravind May 17 '16 at 20:06
  • $\begingroup$ I thought about this possibility, too. But on the blackboard it was $[-L,L]$. Hm, maybe a typo, dont know. $\endgroup$ – H. Hawks May 17 '16 at 21:14

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