MATH528

MATH528 Lesson12: ODE series solution. Special functions

Remark. The real numbers are a complete, ordered field $(ℝ, +, \times)$

Remark. Power series are simply an infinite sequence of the operations defined in $ℝ$

S_{n} (x) = \sum_{j = 0}^{n} a_{j} x^{j}, n \in ℕ

Remark. Power series can also be interpreted as a sequence of scalar products

S_{n} (x) = 𝒂^{T} 𝒃 (x), 𝒂^{T} = (\begin{array}{llll} a_{0} & a_{1} & \dots & a_{n} \end{array}), 𝒃 (x) = {(\begin{array}{llll} 1 & x & \dots & x^{n} \end{array})}^{T}

The power series method to solve ODEs onsist of:

Introducing a representation $y (x) =$ $\sum_{j = 0}^{\infty} a_{j} x^{j}$
Replacing the representation into the ODE of interest and identifying coefficients of powers of $x$

Example. $y^{'} = y$ , $y (0) = y_{0}$ , Try $y = a_{0} + a_{1} x + \dots + a_{n} x^{n} + \dots +$

\begin{array}{rcl} y^{'} = a_{1} + 2 a_{2} x + \dots + (n + 1) a_{n + 1} x^{n} & = & y = a_{0} + a_{1} x + \dots + a_{n} x^{n} + \dots + = y \Rightarrow \\ a_{1} = a_{0}, & a_{2} = \frac{1}{2} a_{1}, \dots & a_{n + 1} = \frac{1}{n + 1} a_{n} \Rightarrow \\ y (x) = & (1 + \frac{1}{1!} x + \frac{1}{2!} x^{2} + \dots) a_{0} & = e^{x} y (0), a_{0} = y (0) . \end{array}