# an eigenvalue problem of a differential equation

Let $L = - (\frac{d}{d \theta})^2$. Consider the eigenvalue problem $L \phi = \lambda \phi \; (a < \theta < b )$. If $\phi(a) = \phi(b) = 0$, can I derive $\lambda_n$ formula ? Here $\lambda_n$ means $n$th eigen value ( $\lambda_1 \leqslant \lambda_2 \leqslant \cdots$).

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Is this homework? What have you tried? Does it help to rephrase in this way: What functions when you take their derivatives twice give a you a multiple of that function again? –  Matt Jul 27 '12 at 1:15
@Matt Yes, infact I derived $\phi = C_1 \cos \sqrt{\lambda} \theta + C_2 \sin \sqrt{\lambda} \theta$, and by the initial condition $\phi(a) = 0$ and $\phi(b) = 0$, I have two equations, but I want to know the next steps.. –  Bamily Jul 27 '12 at 1:20
Find the solutions that satisfy the boundary conditions. –  copper.hat Jul 27 '12 at 1:28
Hint: You may find it somewhat easier if you write $\phi = C_1 \cos(\sqrt{\lambda}(\theta - a)) + C_2 \sin(\sqrt{\lambda}(\theta - a))$. –  Robert Israel Jul 27 '12 at 3:07
@RoberIsrael Thank you Robert, I solved it by your favor. –  Bamily Jul 27 '12 at 3:44