Taylor Series Solution of Initial Value Problems

5 Taylor Series Solution of Initial Value Problems 

In Section 3.5.1, the Frobenius series method was discussed with regard to differential equations with regular singular points. In this section, a method is given that effectively deals with differential equations with ordinary points. To illustrate the importance of this method consider the relatively harmless-looking first-order initial value problem 

This problem has a unique solution (Theorem 2.1) for all x and is given by 

However, the above equation 3.100 is not in closed form. By closed form we mean 

Notice that one can take this process up to any desired amount of terms. Equation 3.104 is more manageable than Equation 3.100. 

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