This terminology is because the solutions correspond to the eigenvalues and eigenfunctions of a Hermitian differential operator in an appropriate function space. Such values of λ, when they exist, are called the eigenvalues of the problem, and the corresponding solutions are the eigenfunctions associated to each λ. The value of λ is not specified in the equation: finding the λ for which there exists a non-trivial solution is part of the given S-L problem. Each such equation ( 1) together with its boundary conditions constitutes a Sturm–Liouville (S-L) problem. (In the case of more general p( x), q( x), w( x), the solutions must be understood in a weak sense.) In addition, y is typically required to satisfy some boundary conditions at a and b. 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 ( a, b) and satisfies the equation ( 1) at every point in ( a, b). All second-order linear ordinary differential equations can be reduced to this form. The function w( x), sometimes denoted r( x), is called the weight or density function. For given coefficient functions p( x), q( x), and w( x) and an unknown function y of the free variable x.
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