Introduction to Series Methods

In Chapter 2 you learned that all linear differential equations with constant coefficients of the type, for example. 

This series representation ean be written in compact form 

We know in advance that a = r /n, but nonetheless such a procedure may be quite useful for cases when an analytieal solution is not available, such as when nonconstant coefficients arise [i.e., a ix), flo( )l- 

Let us proceed to solve Eq. 3.1 by the series representation, Eq. 3.4. Assume that the series coefficients a are unknown and we must hnd them. First, differentiate the series, Eq. 3.4, and insert the several derivatives into the defining Eq. 3.1 to see if useful generalizations can be made. 

Inserting these into Eq. 3.1 yields the necessary condition 

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