Classification of Sturm-Liouville Problems

A general homogeneous second-order differential equation of the form 

Generally, there are three classes of Sturm-Liouville problems  

Each class is discussed below with an illustrative problem. Again the boundary conditions are very influential as to what class of boundary value problem one has to solve. 

If 2 = 0, then we get a trivial solution. Then, for nontrivial solution to exist 

Therefore, A = Ar defines the so-called eigenvalues of this problem. Corresponding to each eigenvalue is an eigenfunction 

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