Self-adjoint form
In mathematics and its applications, a classical Sturm–Liouville theory, named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), is the theory of a real second-order linear differential equation of the form
![{\frac {{\mathrm {d}}}{{\mathrm {d}}x}}\left[p(x){\frac {{\mathrm {d}}y}{{\mathrm {d}}x}}\right]+q(x)y=-\lambda w(x)y,](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b80b2cfbaabababb3a0f02a570c4a705d7e748b)
()
where y is a function of the free variable x. Here the functions p(x), q(x), and w(x) > 0 are specified at the outset. In the simplest of cases all coefficients are continuous on the finite closed interval [a,b], and p has continuous derivative. In this simplest of all cases, this 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). In addition, the unknown 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.
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