Concept

# Sturm–Liouville theory

Summary
In mathematics and its applications, classical Sturm–Liouville theory (developed by Joseph Liouville and Jacques Charles François Sturm) is the theory of real second-order linear ordinary differential equations of the form: for given coefficient functions p(x), q(x), and w(x), an unknown function y=y(x) of the free variable x, and an unknown constant \lambda. All homogeneous (i.e. with the right-hand side equal to zero) second-order linear ordinary differential equations can be reduced to this form. In addition, the solution y is typically required to satisfy some boundary conditions at extreme values of x. Each such equation () together with its boundary conditions constitutes a Sturm–Liouville problem. In the simplest case where all coefficients are continuous on the finite closed interval and p has continuous derivative, a function y = y(x) is called a solution if it is cont
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