Question

Let’s say one is given the ODE y” + ay = 0 with boundary conditions y'(0) = 0 and y'(1) = 0. How would one use Mathematica to find the eigenvalues and eigenfunctions? What if one is given a more general ODE, let’s say y” + (y^2 – 1/2)y = 0 with the same boundary conditions?

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This question has been answered by Simon’s comment below.

Answer 1

DSolve only gives solutions for “generic” parameters, which is why

DSolve[y''[x] + a^2 y[x] == 0 && y'[0] == 0 && y'[1] == 0, y, x]

only returns the trivial {{y -> Function[{x}, 0]}}.

If you’re considering $-a^2$ to be an eigenvalue of the second derivative operator with the 0 velocity boundary conditions, first solve

In[1]:= sol = DSolve[y''[x] + a^2 y[x] == 0, y, x]
Out[1]= {{y -> Function[{x}, C[1] Cos[a x] + C[2] Sin[a x]]}}

then enforce the boundary conditions using Reduce
(where, to simplify the result, I’ve also assumed that a != 0
and that sol is not trivial)

In[2]:= Reduce[y'[0] == 0 && y'[1] == 0 && 
               a != 0 && (C[1] != 0 || C[2] != 0) /. sol, 
               a] // FullSimplify

Out[2]= Element[C[3], Integers] && C[2] == 0 && C[1] != 0 && 
        ((a == 2*Pi*C[3] && a != 0) || Pi + 2*Pi*C[3] == a)

which says that the eigenvectors are proportional to $\cos(a x)$ for $a = 2 n \pi$ or $a = (2 n + 1) \pi$ with $n$ an integer.

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As for the second equation in your question, it only makes sense to talk about eigenvectors for linear operators. For nonlinear differential equations, eigenvectors are useful for examining the linearized behaviour around critical points.