# 7: Series Solutions of Linear Second Order Equations

- Page ID
- 9445

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

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.

- 7.0: 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.

- 7.1: 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.

- 7.2: 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).

- 7.3: 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.

- 7.4: Regular Singular Points Euler Equations
- This section introduces the appropriate assumptions on P₁ and P₂ in the case where P₀(0)=0, and deals with Euler's equation $$ ax^2y''+bxy'+cy=0, \nonumber $$ where a, b, and c are constants. This is the simplest equation that satisfies these assumptions.

- 7.5: The Method of Frobenius I
- In this section we begin to study series solutions of a homogeneous linear second order differential equation with a regular singular point at x0=0, so it can be written as x²A(x)y″+xB(x)y′+C(x)y=0, where A, B, C are polynomials and A(0)≠0.

- 7.6: The Method of Frobenius II
- In this section we discuss a method for finding two linearly independent Frobenius solutions of a homogeneous linear second order equation near a regular singular point in the case where the indicial equation has a repeated real root.

- 7.7: The Method of Frobenius III
- Previously, we discussed methods for finding Frobenius solutions of a homogeneous linear second order equation near a regular singular point in the case where the indicial equation has a repeated root or distinct real roots that do not differ by an integer. In this section we consider the case where the indicial equation has distinct real roots that differ by an integer.