ABSTRACT
Chapter 2 discussed the power series method to find an approximate analytical solution to ordinary differential equations (ODEs). We observed that the series solutions obtained were all convergent in nature, although in some cases, the region of convergence is small. But, the major drawback of it is that the method fails to provide solutions to equations at irregular singular points (IRSPs) and solutions at infinity. Note that, as such, there is no general theory for solution of an ODE about an IRSP (Paris and Wood, 1986). However, the method presented in this chapter-the asymptotic method-provides solutions to problems with irregular singularity at infinity. This method can also be used to provide solutions at infinity to differential equations and to the so-called singularly perturbed problems, which are discussed in detail in Chapter 4. This chapter is limited to the applications of this method to linear differential equations. Refer to the work of Bayat et al. (2012) for applications of the method to nonlinear differential equations. This chapter also provides a method for solving equations involving large parameters.
