ORDINARY POINT SERIES SOLUTIONS

Consider the second order differential equation

P(x)y'' + Q(x)y' + R(x)y = 0

Ordinary and Singular Points

• A singular point is a point x₀ where P(x₀) = 0.
• An ordinary point is a point x₀ where $P(x_0) \neq 0$.

Solutions Near an Ordinary Point at x₀

• Use $y(x) = \sum\limits_{n=0}^\infty a_n(x-x_0)^n .$
• Determine series for P(x), Q(x) and R(x).
• Substitute series for y, P, Q and R into equation.
• Use series algebra to form a single series.
• Find a recurrence relation for the a_n's.

Taylor Series

• If $P(x_0) \neq 0$, consider y'' + p(x)y' + q(x)y = 0,
  where p(x)=Q(x)/P(x) and q(x)=R(x)/P(x).
• Let $p_0=p(x_0), p'_0=p'(x_0), \ldots$, then
  
  $$\begin{eqnarray*}y''(x_0) &=& -q_0y(x_0) - p_0y'(x_0)\\ y'''(x_0) &=&(p_0q_0-q'... ...''(x_0)}{2!}(x-x_0)^2\cdots \\ &=& y(x_0)y_1(x)+y'(x_0)y_2(x). \end{eqnarray*}$$
  
  with nonzero radius of convergence for series.
• Lower bound for radius of convergence $\rho$ is minimum
  of radii of convergence for p(x) and q(x); $\rho$ is distance
  from x₀ to nearest singularity of p(x), q(x).