tailieunhanh - Ebook Mathematical methods for physics and engineering (3/E): Part 2

Part 2 book “Mathematical methods for physics and engineering” has contents: Integral equations, complex variables, numerical methods, group theory, representation theory, quantum operators, applications of complex variables, calculus of variations, and other contents. | 17 Eigenfunction methods for differential equations In the previous three chapters we dealt with the solution of differential equations of order n by two methods. In one method, we found n independent solutions of the equation and then combined them, weighted with coefficients determined by the boundary conditions; in the other we found solutions in terms of series whose coefficients were related by (in general) an n-term recurrence relation and thence fixed by the boundary conditions. For both approaches the linearity of the equation was an important or essential factor in the utility of the method, and in this chapter our aim will be to exploit the superposition properties of linear differential equations even further. We will be concerned with the solution of equations of the inhomogeneous form Ly(x) = f(x), () where f(x) is a prescribed or general function and the boundary conditions to be satisfied by the solution y = y(x), for example at the limits x = a and x = b, are given. The expression Ly(x) stands for a linear differential operator L acting upon the function y(x). In general, unless f(x) is both known and simple, it will not be possible to find particular integrals of (), even if complementary functions can be found that satisfy Ly = 0. The idea is therefore to exploit the linearity of L by building up the required solution y(x) as a superposition, generally containing an infinite number of terms, of some set of functions {yi (x)} that each individually satisfy the boundary conditions. Clearly this brings in a quite considerable complication but since, within reason, we may select the set of functions to suit ourselves, we can obtain sizeable compensation for this complication. Indeed, if the set chosen is one containing functions that, when acted upon by L, produce particularly simple results then we can ‘show a profit’ on the operation. In particular, if the 554 EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS set consists of those functions yi for