ABSTRACT
This chapter deals with the case when the engineer starts from a differential equation with certain boundary conditions that is difficult to solve. It discusses the variational form of eigenvalue problems and the Sturm-Liouville class of differential equations. Executing the inverse of the Euler-Lagrange process and obtaining the variational formulation of the boundary value problem may be advantageous. Eigenvalue problems of various kinds may also be formulated as variational problems. The eigenvalue problem has an infinite number of eigenvalues, and for each eigenvalue there exists a corresponding eigenfunction that is unique apart from a constant factor. Hence, the variational form should also provide means for the solution of multiple pairs. The subsequent eigensolutions may be found by the same procedure and the sequence of the eigenpairs attain the extrema of the variational problem under the successive conditions of the orthogonality against the preceding solutions.
