What (non-trivial) functions satisfy $f(a)=f(b)=f'(a)=f'(b)=0$? I am working on some undergrad research and would like to understand this a bit more.
Context
I am investigating a differential equation of the form $y^{(4)}+ky=0$, where $k$ is a parameter. Basically an Sturm-Liouville problem of 4th order.
Any function of the form $f(x) = (x-a)^2(x-b)^2 g(x)$, where $g$ is differentiable, satisfies those boundary conditions. That's a pretty big family.
Also, any linear space $V$ of differentiable function such that $\dim V\ge 5$ contains nontrivial functions that satisfy the conditions. Indeed, the zero set of a linear functional has codimension $1$, and the common zero set of $4$ linear functionals has codimension $4$.
For example, the space of functions of the form $\sum_{k=1}^5 c_k e^{\lambda_k x}$ contains some such functions.
However the space of solutions of $y^{(4)}+ky=0$ is only $4$-dimensional. So we should expect that normally, there will not be any nontrivial solutions. Of course, we might get lucky: when $k$ is a specific constant (depending on $b-a$), there will be solutions. You can dig into this by writing down the general solution $y(x) = \sum_{k=1}^4 c_4 e^{\lambda_k x}$ and plugging this into the boundary conditions. Some $\lambda_k$ are complex here; trigonometric functions can be used to deal with those.