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

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

Frobenius Solutions¶

for second linear ODE

Ordinary Points¶

$x_0$ is an ordinary point if $P(x_0)\ne 0$, 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¶

• r may be negative or even non-integer

• notice that

summation start from $n=0$ instead of $n=1$

Frobenius Methods¶

the second order ODE

which $x_0$ is a Regular Singular Points

by solving it with

we can get two root $r_1,r_2$

• Case $|r_1-r_2|\notin \mathbb{N}$
  • 2 linearly independent Frobenius solutions
  • $y_1(x)=\sum c_n(x-x_0{)}^{n+r_1}$
  • $y_2(x)=\sum c_n^{\prime }(x-x_0{)}^{n+r_2}$
  • $c_0,c_0^{\prime }\ne 0$

• Case $r_1=r_2$

  • 2 linearly independent Frobenius solutions
  • $y_1(x)=\sum c_n(x-x_0{)}^{n+r_1}$
  • $y_2(x)=y_1(x)\cdot \ln (x)+\sum c_n^{\prime }(x-x_0{)}^{n+r_1}$
  • $c_0,c_0^{\prime }\ne 0$

• Case $|r_1-r_2|\in \mathbb{N}$

  • $y_1(x)=\sum c_n(x-x_0{)}^{n+r_1}$
  • $y_2(x)=k{y}_1(x)\ln (x)+\sum c_n^{\prime }(x-x_0{)}^{n+r_2}$
  • $c_0,c_0^{\prime }\ne 0$