Let $a>0$ be a constant positive number. I am stuck trying to solve the following regular Sturm-Liouville problem:
$$\frac{\mathrm d}{\mathrm d x}((a+x)f'(x)) = -v f(x),\qquad f(0)=f(1)=0$$
where $v$ is the eigenvalue.
According to Mathematica, the general solution of the ODE is (found by running DSolve[{D[(x + a) f'[x], x] + v f[x] == 0}, f[x], x]
):
$$f(x) = c_1 I_0(2\sqrt{-(a+x)v}) + c_2 K_0(2\sqrt{-(a+x)v})$$
However, no combination of $c_1,c_2,v$ can give a non-trivial function $f(x) \not\equiv 0$ while satisfying $f(0)=f(1)=0$. However according to Sturm Liouville theory, eigenvalues and non-trivial eigenfunctions must exist.
So how can I use Mathematica to solve this problem? Or, how can I obtain the general real solution of an ODE with real coefficients?
DSolve[{y''[x] + b ^2 y[x] == 0, y[0] == 0, y[1] == 0}, y[x], x]
correctly solves the eigenvalue problem, without having to separate the b.c.s. $\endgroup$ – becko Jul 27 '17 at 20:08