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Matakuliah Tahun : K0124 / Matematika Teknik II : 2006/2007 PERTEMUAN 8 STRUM-LIOVILLE SYSTEMS 1 A boundry value problem having the form d dy p ( x ) q ( x) r ( x)y 0 dx dx axb a1 y (a ) a 2 y ' (a) 0, b1 y (b) b2 y ' (b) 0 2 where a1 , a2 , b1 , b2 are given constants; p( x), q( x), r ( x) are given functions which we shall assume to be differentiable and is an unspecified parameter independent of x, is called a Sturm-Liouville boundry-value problem or Sturm-Liouville System. 3 A non-trivial solution of this system, i.e. one which is not identically zero, exist in general only for a particular set of values of the parameter . These values are called the characteristic values, or more often eigenvalues, of the system. 4 The corresponding solution are called characteristic functions or eigenfunctions of the system. In general to each eigenvalue there is aone eigenfunction, although exceptions can occur. 5 If p(x), q(x) are real, then the eigenvalues are real. Also the eigenfunctions form an orthogonal set with respect to the density function r(x) which is generally taken as non-negative , i.e. r(x) 0. 6 It follows that by suitable normalization the set of functions can be made an orthonormal set with respect to r(x) in axb . 7 TERIMA KASIH 8