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In mathematics and its applications, a classical Sturm–Liouville equation, named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), is a real second-order linear differential equation of the form where y is a function of the free variable x. Here the functions p(x) > 0 has a continuous derivative, q(x), and w(x) > 0 are specified at the outset, and in the simplest of cases are continuous on the finite closed interval [a,b]. In addition, the function y is typically required to satisfy some boundary conditions at a and b. The function w(x), which is sometimes called r(x), is called the "weight" or "density" function. The value of λ is not specified in the equation; finding the values of λ for which there exists a non-trivial solution of (1) satisfying the boundary conditions is part of the problem called the Sturm–Liouville problem (S L). Such values of λ when they exist are called the eigenvalues of the boundary value problem defined by (1) and the prescribed set of boundary conditions. The corresponding solutions (for such a λ) are the eigenfunctions of this problem. Under normal assumptions on the coefficient functions p(x), q(x), and w(x) above, they induce a Hermitian differential operator in some function space defined by boundary conditions. The resulting theory of the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their completeness in a suitable function space became known as Sturm–Liouville theory. This theory is important in applied mathematics, where S–L problems occur very commonly, particularly when dealing with linear partial differential equations that are separable.