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I have to check if I can get the eigenvalues for a specific equation. Actually I just want the first 2 beta:

eqy = Y''[y] + \[Beta]^2*Y[y] == 0;

So I've tried to write the boundary conditions and the DSolve:

ic1 = Y'[y] == 0 /. y -> 0;
ic2 = Y'[y] == 0 /. y -> 0.5;
sol = DSolve[{eqy}, ic1, ic2, Y[y], {y, 0, 0.5}]

Unlikely this results in "Y'[0]==0 cannot be used as a function. I don't know how to solve it. The result should be: enter image description here

with the following as (22) and (23). Psi is a function with separable variables: Psi=X(x)*Y(y)*G(t). enter image description here

Thank you in advance for your help.

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    $\begingroup$ Your syntax is wrong, it should read: ?sol = DSolve[{eqy, ic1, ic2}, Y[y], {y, 0, 0.5}] $\endgroup$
    – Daniel Huber
    Mar 9, 2021 at 9:36
  • $\begingroup$ I'm very sorry! I made such a stupid mistake. Thank you so much!!! $\endgroup$
    – Roberta Mongillo
    Mar 9, 2021 at 9:45

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