6: Series Solutions of Linear Second Order Equations

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In this Chapter, we study a class of second order differential equations that occur in many applications, but cannot be solved in closed form in terms of elementary functions.

6.1: Prelude to Series Solutions of Linear Second Order Equations
Second order differential equations occur in many applications, but cannot be solved in closed form in terms of elementary functions including Bessel's, Airy's and Langendre's Equations that can be written in the form P0(x)y′′+P1(x)y′+P2(x)y=0. These equations do not, in general, have closed form solutions, we seek series representations for solutions.
6.2: Review of Power Series
Many applications give rise to differential equations with solutions that can’t be expressed in terms of elementary functions such as polynomials, rational functions, exponential and logarithmic functions, and trigonometric functions. The solutions of some of the most important of these equations can be expressed in terms of power series. We’ll study such equations in this chapter. In this section we review relevant properties of power series.
- 6.2.1: Review of Power Series (Exercises)
6.3: Series Solutions Near an Ordinary Point I
This section is devoted to finding power series solutions of (A) in the case where $P_0(0)~\ne~0$. The situation is more complicated if P_0(0)=0; however, if P_1 and P_2 satisfy assumptions that apply to most equations of interest, then we're able to use a modified series method to obtain solutions of (A).
- 6.3.1: Series Solutions Near an Ordinary Point I (Exercises)
6.4: Series Solutions Near an Ordinary Point II
In this section we continue to find series solutions of initial value problems . For the equations considered here it is difficult or impossible to obtain an explicit formula for an in terms of n . Nevertheless, we can calculate as many coefficients as we wish. The next three examples illustrate this.
- 6.4.1: Series Solutions Near an Ordinary Point II (Exercises)