ODE Lecture Notes
Section 5.3
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Section 5.3: Series Solutions Near an Ordinary Point, Part II
Big Idea: The power series solution technique of solving a second-order differential equation is
on a firm theoretical footing. That theory gives us insight into the radius of convergence of the
solution based on the original differential equation.
Big Skill: You should be able to compute the minimum radius of convergence of a power series
solution.
Note:
To prove a solution of P  x  y  Q  x  y  R  x  y  0  y  p  x  y  q  x  y can be written as

y    x    an  x  x0  , we must be able to compute m!am   
n
n 0
m
 x0  for any order derivative
m based only on the information given by the IVP.
Practice:
1. Determine the initial value for first four derivatives of the solution of the IVP
x2 y  1  x  y  3ln  x  y  0 , y(1) = 2, y(1) = 0.
ODE Lecture Notes
Section 5.3
Page 2 of 2
Note:
So it seems that p and q need to at least be infinitely differentiable at x0, but in addition they also
need to be analytic at x0.
Theorem 5.3.1
If x0 is an ordinary point of the differential equation P  x  y  Q  x  y  R  x  y  0 (i.e.,
pQ
P
and q  R

P
are analytic at x0), then its general solution is
y   an  x  x0   a0 y1  x   a1 y2  x  ,
n
n 0
where a0 and a1 are arbitrary, and y1 and y2 are two power series solutions that are analytic at x0.
The solutions y1 and y2 form a fundamental set of solutions. Also, the radius of convergence of
y1 and y2 is at least as large as the minimum of the radii of convergence of p and q.
Note:
From the theory of complex-valued rational expressions, it turns out that radius of convergence
of a power series of a rational expression about a point x0 is the distance from x0 to the nearest
zero of the (complex-valued) denominator.
Practice:
2. Determine a lower bound for the radius of convergence of the solution of the Legendre
equation 1  x 2  y  2 xy     1 y  0 about x0 = 0.
3. Determine a lower bound for the radius of convergence of the solution of the Legendre
equation 1  x 2  y  2 xy  4 x 2 y  0 about x0 = 0 and x0 = ½.