8.1 Power Series Solutions

Power series solutions are a powerful tool for solving differential equations when other methods fall short. They allow us to represent solutions as infinite sums, providing a way to analyze behavior near singular points.

In this section, we'll learn how to construct power series solutions, determine their convergence, and classify singular points. We'll also explore techniques like recurrence relations and the indicial equation to solve differential equations using power series.

Power Series and Convergence

Fundamentals of Power Series

• Power series express functions as an infinite sum of terms involving powers of a variable $x$ multiplied by coefficients $a_n$
  • General form: $\sum_{n=0}^{\infty} a_n (x - c)^n = a_0 + a_1(x - c) + a_2(x - c)^2 + \cdots$
  • $c$ represents the center of the power series
• Radius of convergence $R$ determines the interval on which a power series converges
  • Inside the interval $(c - R, c + R)$, the series converges
  • Outside this interval, the series diverges
  • On the boundary points $x = c \pm R$, further investigation is needed
• Ordinary points of a differential equation are where the coefficients of the equation are analytic (can be represented by a convergent power series)

Classifying Singular Points

• Regular singular points of a differential equation are where the coefficients have a pole of order at most 1
  • The point $x = 0$ is a regular singular point if the equation can be written as $x^2 y'' + xp(x)y' + q(x)y = 0$, where $p(x)$ and $q(x)$ are analytic at $x = 0$
  • Solutions near regular singular points can be represented by power series
• Irregular singular points are where the coefficients have a pole of order greater than 1 or essential singularity
  • Power series solutions may not exist at irregular singular points

Solving Power Series

Recurrence Relation and Indicial Equation

• Recurrence relations are equations that relate each coefficient $a_n$ to previous coefficients in a power series solution
  • Obtained by substituting the power series into the differential equation and equating coefficients of like powers of $x$
  • Example: For the equation $xy'' + y' - xy = 0$, the recurrence relation is $a_{n+2} = \frac{n - 1}{(n + 1)(n + 2)} a_n$
• The indicial equation determines the possible values of the power series exponent at a regular singular point
  • Found by substituting $y = x^r$ into the differential equation and solving for the lowest power of $x$
  • Roots of the indicial equation give the values of $r$, which determine the leading terms of the series solutions

Solving Techniques

• The method of undetermined coefficients is used to solve linear differential equations with power series
  • Assume a solution of the form $y = \sum_{n=0}^{\infty} a_n x^n$
  • Substitute the series into the differential equation and solve for the coefficients $a_n$ using the recurrence relation
• Taylor series solutions expand a function about a point $x = c$ using its derivatives
  • The coefficients are given by $a_n = \frac{f^{(n)}(c)}{n!}$, where $f^{(n)}$ denotes the $n$-th derivative of $f$
  • Example: The Taylor series for $e^x$ about $x = 0$ is $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$