1 Conditions for using BVP eigenvalue method

We are only here talking about second order linear ode of the form \(a\left ( x\right ) y^{\prime \prime }+b\left ( x\right ) y^{\prime }+c\left ( x\right ) y=0\) with boundary conditions. If it is an IVP ode, then this note does not apply. The first thing to check for is that B.C. are homogeneous. Which means they are equal to zero. An example is \(y\left ( 0\right ) =0,y\left ( L\right ) =0\) or \(y\left ( 0\right ) =0,y^{\prime }\left ( L\right ) =0\) and so on. The BC do not have to be in terms of unknown \(L\). They can be \(y\left ( 0\right ) =0,y\left ( 1\right ) =0\) for example. If the BC are not homogeneous then we have to do some preprocessing which is not considered here.

There are two general cases. The first is if the ode itself has an unknown in it or not. For example \(y^{\prime \prime }+y^{\prime }+\lambda y=0\) would qualify. Also \(ay^{\prime \prime }+y^{\prime }+y=0\) will qualify. The second general case is when the ode has no unknown it is. Which means all the coefficients are known. Hence there is a total of 4 cases to consider:

1. The ode has at least one unknown in it such as \(y^{\prime \prime }+y^{\prime }+\lambda y=0\) but the BC have no unknown. For example \(y\left ( 0\right ) =0,y\left ( 1\right ) =0\). This is a BVP eigenvalue problem. The above gives the solution

   \[ y\left ( x\right ) =\left \{ \begin {array} [c]{ccc}2e^{-\frac {x}{2}}c_{1}\sinh \left ( \frac {1}{2}\sqrt {1-4\lambda }x\right ) & & \sinh \left ( \frac {1}{2}\sqrt {1-4\lambda }\right ) =0\\ 0 & & \text {otherwise}\end {array} \right . \]

2. The ode has at least one unknown in it such as \(y^{\prime \prime }+y^{\prime }+\lambda y=0\) and also BC have unknown. For example \(y\left ( 0\right ) =0,y\left ( L\right ) =0\) where here \(L\) is unknown. This is also BVP eigenvalue problem

   \[ y\left ( x\right ) =\left \{ \begin {array} [c]{ccc}2e^{-\frac {x}{2}}c_{1}\sinh \left ( \frac {1}{2}\sqrt {1-4\lambda }x\right ) & & \sinh \left ( \frac {1}{2}\sqrt {1-4\lambda }L\right ) =0\\ 0 & & \text {otherwise}\end {array} \right . \]

3. The ode has no unknown in it such as \(y^{\prime \prime }+y^{\prime }+5y=0\) but BC have at least one unknown. For example \(y\left ( 0\right ) =0,y\left ( L\right ) =0\) where here \(L\) is unknown. This is BVP but not an eigenvalue problem. There is one eigenfunction The solution will have the unknown \(L\) in it.

   \[ y\left ( x\right ) =\left \{ \begin {array} [c]{ccc}e^{-\frac {x}{2}}c_{1}\sin \left ( \frac {\sqrt {19}}{2}x\right ) & & L=\frac {4n\pi }{\sqrt {19}}\text {or\ }L=\frac {2\left ( \pi +2n\pi \right ) }{\sqrt {19}}\text { where }n\in \mathbb {Z} \\ 0 & & \text {otherwise}\end {array} \right . \]
4. The ode has no unknown in it such as \(y^{\prime \prime }+y^{\prime }+5y=0\) and BC also has no unknown. For example \(y\left ( 0\right ) =0,y\left ( 1\right ) =0\). This is BVP but not an eigenvalue problem. The solution  is \(y\left ( x\right ) =0\) in this case.

In the above, only 1,2 are considered BVP eigenvalue problems. The others are just BVP problems. In (3), even though there is no eigenvalues (because the ODE has no unknown in it), we still give conditions for non-trivial solution, depending on value of \(L\). In (4), only trivial solution is possible.