Why can't a second order differential equation have 2 Neuman conditions? I read that a problem such as:
\begin{equation}
    \begin{cases}
       -\frac{\partial^2 y}{\partial x^2} + y = f(x)\\
       \mathrm{condition} \ n°1\\
       \mathrm{condition}  \ n°2
    \end{cases}
\end{equation}
Could have a unique solution if $\mathrm{condition} \ 1$ and $\mathrm{condition} \ 2$ were:

*

*Initial conditions (Cauchy problem)

*2 Dirichlet conditions ($y(a) = y_a, \ y(b) = y_b \ | \ b > a, \ x \in [a, b]$)

*1 Dirichlet condition + 1 Neuman condition ($y'(a) = y_a, \ y(b) = y_b \ | \ b > a, \ x \in [a, b]$ or the contrary)

Why can't we have 2 Neuman conditions ?

Think about a beam:

*

*2 initial conditions mean you know were the left side of the beam preciselly is and what is its direction (e. g. it's in the wall, so the beam outside is // to the ground)

*2 Dirichlet conditions mean you know where the 2 sides of the beam are (e. g. the beam is between two tables, and you seat in the middle of the beam)

*1 Dirichlet and 1 Neuman mean you know for instance that one side of the beam is on a table, but the other one in a "moving wall"

2 Neuman would mean that both of the beam's sides are in the "moving walls", which can't have a unique solution, since the distance between the 2 sides in the walls can take any value you want. You have thus an infinity of solutions.
