ABSTRACT
It is often the case that the engineer starts from a differential equation with certain boundary conditions, which is difficult to solve. Executing the inverse of the Euler-Lagrange process and obtaining the variational formulation of the boundary value problem may also be advantageous. For differential equations, partial or ordinary, containing a linear, self-adjoint, positive operator, the task may be accomplished. This chapter demonstrates the inverse process through the example of Poisson's equation, a topic of much interest for engineers. Eigenvalue problems of various kinds may also be formulated as variational problems. The eigenvalue problem has an infinite sequence of eigenvalues and for each eigenvalue there exists a corresponding eigensolution that is unique apart from a constant factor. Hence, the variational form should also provide means for the solution of multiple pairs. The chapter also discusses Sturm-Liouville differential equations.
