Power Series solution
In case nothing works out for solving differential equations. This is used as a last resort to find a solution.
Analytic Function #
A function f(x) is said to be an analytic function about point x=x₀ if we can write taylor series expansion about x₀.
Ordinary point of D.E. #
A point x=x₀ is said to be ordinary point of $y’’+P(x)y’+Q(x)y=0$ if P,Q are both analytic at x₀
Singular point of D.E. #
If the point is not ordinary it is singular.
Regular singularity #
The following 2 limits must exist and be finite.
$$ \lim {x\rightarrow x{0}} (x-x_{0})P(x) $$ $$ \lim {x\rightarrow x{0}} (x-x_{0})^2Q(x) $$
Or we can check if both those expressions are analytic at $x_0$
Irregular singularity #
If x₀ is not a , then it’s an irregular singularity.
Power series #
Power series solution at ordinary point.
If x₀ is an ordinary point of the D.E. and has two non trivial linearly independent solutions then we consider power series as a trial solution.
$$ \begin{align*} y=\sum\limits_{n=0}^{\infty} a_{n}(x-x_{0})^{n}\\
a_{n}=\frac{D^{n}y(x_{0})}{n!}\\ Radius\\ of\\ convergence:\\ |x-x_{0}|<\lim {n\rightarrow \infty} \frac{a{n}}{a_{n+1}} \end{align*} $$
We find y’ and y’’ and substitute it in the D.E. to find the coefficients of the power series.
$$ \begin{align*} y’&= \sum\limits_{n=1}^{\infty} na_{n}(x-x_{0})^{n-1}\\ y’’ &= \sum\limits_{n=2}^{\infty} n(n-1) a_{n}(x-x_{0})^{n-2} \end{align*} $$