## differential equations – Euler Buckling using DEigensystem

I’m new to using the functions DEigensystem and DEigenvalue. I started with the examples given in the Documentation system, and it works fine. I decided to test it out on the classical Euler buckling problem – in particular, the more general form, where the governing differential equation is fourth order.

I start with the equation of the elastica parameterized by the angle along the arc length, linearize it, and then use the small slope approximation to write it in terms of deflection in Cartesian coordinates:

``````elastica = (Theta)''(s) + (Lambda)^2 Sin((Theta)(s));
elasticaLin = Series(elastica, {(Theta)(s), 0, 1}) // Normal
elasticaLinW = elasticaLin /. {(Theta)(s) -> w''(x), (Theta)''(s) -> w''''(x)}
``````

This gives:
$$lambda^2 w”(x) + w^{(4)}(x)$$

Of course, this equation can be obtained in a more “Mathematica” way by:

``````(ScriptCapitalL) = Laplacian(Laplacian(w(x), {x}), {x}) + (Lambda)^2 Laplacian(w(x), {x})
``````

I defined the Dirichlet boundary conditions of $$w(0)==w(L)==0$$ as:

``````(ScriptCapitalB)1 = DirichletCondition(w(0) == 0, True);
(ScriptCapitalB)2 = DirichletCondition(w(L) == 0, True);
``````

Finally, trying to use `DEigensystem` to find the first few eigenfunctions, I use the following code, which Mathematica just returns to me when I run it:

``````DEigensystem({(ScriptCapitalL), (ScriptCapitalB)1, (ScriptCapitalB)2}, w(x), {x, 0, L},4)
``````

What am I doing wrong here?