I'm currently studying the Dirichlet and Neumann Laplacians, How can I satisfy both boundary conditions, one at each end.

i.e. if I had $$-f''(x)=\lambda f(x), \qquad x \in [-a,a]$$ where $f(-a)=f'(a)=0$


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Using the usual prescription: Solve the equation $-f''(x)=\lambda f(x)$ for each value of $\lambda$ (separate into zero, positive and negative) and substitute your conditions $f(-a)=f'(a)=0$ to check whether that particular value of $\lambda$ gives a nonzero solution.

  • $\begingroup$ could you expand? ... I solved this to get $c_1\exp(x)$ but I feel this is incorrect as $f(-a)=f'(a)=0$ isn't satisfied $\endgroup$ – smith Jan 23 '16 at 23:18
  • 1
    $\begingroup$ Sorry, it is your job to make the computations. Example: for $\lambda=0$ you get $f(x)=c_1x+c_2$ and from the conditions you get $f(-a)=-c_1a+c_2=0$ and $f'(a)=c_1=0$. Etc. $\endgroup$ – John B Jan 23 '16 at 23:19

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