19 Power Series

Power Series¶

An infinite series in \(x\) of the form

\[ \sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots + a_r x^r + \dots \]

where \(\{ a_0, a_1, a_2, \dots \}\) are constants

equation is convergent only when \(x \to 0\)

Non-Algebraic Elementary Functions¶

Transcendental means non-algebraic

Function	Power Series	Intuition
\(e^x\)	\(1 + \frac{x}{1!} + \frac{x^2}{2!} + \dots\)
\(\cos x\)	\(1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \dots\)	Even function so even numbers
\(\sin x\)	\(x - \frac{x^3}{3!} + \frac{x^5}{5!} + \dots\)	Odd function so odd numbers
\(\cosh x\)	\(1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \dots\)
\(\sinh x\)	\(x + \frac{x^3}{3!} + \frac{x^5}{5!} + \dots\)
\(\log(1+x)\)	\(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots\)
\((1+x)^{-1}\)	\(1 - x + x^2 - x^3 + \dots\)
\((1-x)^{-1}\)	\(1 + x + x^2 + x^3 + \dots\)

Solving¶

\[ \begin{aligned} y &= \sum_{n=0}^\infty a_n x^n &&= a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \dots \\ y' &= \sum_{n=1}^\infty a_n n x^{n-1} &&= a_1 + 2a_2 x + 3a_3 x^2 + \dots \\ y'' &= \sum_{n=2}^\infty a_n n(n-1) x^{n-2} &&= 2a_2 + (3 \cdot 2) a_3 x + \dots \\ \end{aligned} \]

Comparing Coefficients¶

By changing index and re-arranging terms, we have to make the following equal

counter start
\(x\) power

2^nd Order¶

Power series solution is only possible if \(x = 0\) is an ordinary point of the DE.

Types of Points¶

Consider a general 2^nd order differential equation with polynomials \(P_1, P_2, P_3\).

\[ \begin{aligned} P y'' + Q y' + R y &= 0 \\ \implies y'' + \frac{Q}{P} y' + \frac{R}{P} y &= 0 \end{aligned} \]

Types	\(P(a)\)
Ordinary Point	\(\ne 0\)
Singular Point	\(= 0\)

Ordinary Point¶

\(x=a\) is an ordinary point of DE equation, if \(P(a) \ne 0\).

Power Series Solution¶

The power series solution of equation is given by

\[ y = \sum_{n=0}^{\infty} a_n x^n \]

General Solution¶

Solving equation as

\[ y = a(\text{PS}_1) + b(\text{PS}_2) \]

where

PS₁ and PS₂ are congruent and linearly-independent power series, for \(x \to 0\)
\(a\) and \(b\) are arbitrary constants

Singular Points¶

Consider limits

\[ \begin{aligned} p&= \lim_{x \to a} (x-a) &\frac{Q(x)}{P(x)}\\ q&= \lim_{x \to a} (x-a)^{\textcolor{hotpink}{2}} &\frac{R(x)}{P(x)} \end{aligned} \]

Both limits exist	Point Type
✅	Regular
❌	Irregular

Frobenius Series Method¶

Differential equations with regular singular points at \(x=0\) can be solved using a power series of the form

\[ \begin{aligned} y &= x^m \sum_{n=0}^\infty a_n x^n \\ &= \sum_{n=0}^\infty a_n x^{m+n} \end{aligned} \]

where \(m\) is constant coefficient called as root/indical/initial value. This is singular points (to be calculated).

Trick to find indical value¶

\[ m(m-1) + p m + q = 0 \]

where \(p\) and \(q\) are limits from equation

Last Updated: 2023-01-25 ; Contributors: AhmedThahir