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In mathematics and its applications, classical Sturm–Liouville theory is the theory of real second-order linear ordinary differential equations of the form:
for given coefficient functions p, q, and w and an unknown function y of the free variable x. The function w, sometimes denoted r, is called the weight or density function.
All homogeneous second-order linear ordinary differential equations can be reduced to this form.
In the simplest case where all coefficients are continuous on the finite closed interval and p has continuous derivative, a function y is called a solution if it is continuously differentiable on and satisfies the equation at every point in. , q, w, the solutions must be understood in a weak sense.] In addition, y is typically required to satisfy some boundary conditions at a and b. Each such equation together with its boundary conditions constitutes a Sturm–Liouville problem.