Essentials of Mathematical Methods in Science and Engineering, 2nd Edition

CHAPTER 12SECOND‐ORDER DIFFERENTIAL EQUATIONS AND SPECIAL FUNCTIONS

In applications, differential equations are usually accompanied by boundary or initial conditions. In the previous chapter, we have concentrated on techniques for finding analytic solutions to ordinary differential equations. We have basically assumed that boundary conditions can in principle be satisfied by a suitable choice of the integration constants in the solution. The general solution of a second‐order ordinary differential equation contains two arbitrary constants, which requires two boundary conditions for their determination. The needed information is usually supplied either by giving the value of the solution and its derivative at some point or by giving the value of the solution at two different points. As in chaotic processes, where the system exhibits instabilities with respect to initial conditions, the effect of boundary conditions on the final result can be drastic. In this chapter, we discuss three of the most frequently encountered second‐order ordinary differential equations of physics, that is, Legendre, Laguerre, and Hermite equations. We approach these equations from the point of view of the Frobenius method and discuss their solutions in detail. We show that the boundary conditions impose severe restrictions on not just the integration constants but also on the parameters that the differential equation itself includes. Restrictions on such parameters may have rather dramatic effects like ...