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Series Solution

Power Series Solution

Analytic

A function \(f(x)\) is analytic at \(x_0\) if \(f(x)\) has a power series representation in some open intervals about \(x_0\), that is

\[ \begin{gather} f(x) = \sum_{n=0}^{\infty}{a_n\ (x-x_0)^n} \quad \text{ for } x \in (x_0-h, x_0+h) \end{gather} \]

where \(a_n\) are the Taylor coefficients of \(f(x)\) at \(x_0\) :

\[ \begin{gather} a_n=\frac{1}{ n!}f^{(n)}(x_0) \end{gather} \]

Frobenius Solutions

for second linear ODE

\[ \begin{gather} P(x)y'' + Q(x)y' + R(x)y=F(x) \\\\ \text{or} \\\\ y''+p(x)y'+q(x)y=f(x) \end{gather} \]

Ordinary Points

\(x_0\) is an ordinary point if \(P(x_0) \neq 0\), and \(\frac{Q}{P}\), \(\frac{R}{P}\), and \(\frac{F}{P}\) are analytic at \(x_0\).

Singular Points

\(x_0\) is a singular point if \(x_0\) is not an ordinary point.

Regular Singular Points

\(x_0\) is a regular singular point if \(x_0\) is a singular point and the function \((x-x_0)\frac{Q}{P}\) and \((x-x_0)^2\frac{R}{P}\) are analytic at \(x_0\).

Irregular Singular Points

A singular point \(x_0\) is a irregular point if it's not a regular singular points.

Frobenius Series

\[ \begin{gather} y(x)=\sum_{n=0}^{\infty}{c_n(x-x_0)^{n+r}} \end{gather} \]

  • r may be negative or even non-integer

  • notice that

\[ \begin{gather} y' = \sum_{n=\bf{0}}^{\infty}c_n(n+r)(x-x_0)^{n+r-1} \end{gather} \]

summation start from \(n=0\) instead of \(n=1\)

Frobenius Methods

the second order ODE

\[ \begin{gather} P\ y'' + Q\ y' + R\ y=F(x) \end{gather} \]

which \(x_0\) is a Regular Singular Points

by solving it with

\[y(x)=\sum_{n=0}^{\infty}{c_n(x-x_0)^{n+r}}\]

we can get two root \(r_1, r_2\)

  • Case \(|r_1-r_2| \notin ℕ\)
    • 2 linearly independent Frobenius solutions
    • \(y_1(x)=\sum{c_n(x-x_0)^{n+r_1}}\)
    • \(y_2(x) =\sum{c'_n(x-x_0)^{n+r_2}}\)
    • \(c_0,\ c'_0 \neq 0\)

  • Case \(r_1 = r_2\)

    • 2 linearly independent Frobenius solutions
    • \(y_1(x)=\sum{c_n(x-x_0)^{n+r_1}}\)
    • \(y_2(x) = y_1(x)\cdot \ln{(x)} + \sum{c'_n(x-x_0)^{n+r_1}}\)
    • \(c_0,\ c'_0 \neq 0\)

  • Case \(|r_1 - r_2| \in ℕ\)

    • \(y_1(x)=\sum{c_n(x-x_0)^{n+r_1}}\)
    • \(y_2(x) = k\ y_1(x)\ \ln{(x)}+\sum{c'_n(x-x_0)^{n+r_2}}\)
    • \(c_0,\ c'_0 \neq 0\)