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Sturm-Liouville theory

In mathematics and its applications, classical Sturm–Liouville theory is the theory of real second-order linear ordinary differential equations of the form : d/dx[p(x)dy/dx] + q(x)y = -λ w(x)y (1), 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 λ. 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 [a, b] and p has continuous derivative, a function y = y(x) is called a solution if it is continuously differentiable and satisfies the equation (1) at every x ∈ ( a , b ). In the case of more general p(x), q(x), w(x), the solutions must be understood in a weak sense. (Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory)

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