1
Series Solutions Near Ordinary Points
A general homogeneous second order linear differential equation with analytic
coefficients (i.e. the functions can be represented by power series) has the form:
𝐴(𝑥 )𝑦 ′′ + 𝐵(𝑥 )𝑦 ′ + 𝐶 (𝑥 )𝑦 = 0.
If 𝐴(𝑥 ) ≠ 0 for any 𝑥 in an open interval, 𝐼, then we can divide the equation by
𝐴(𝑥 ) to get:
𝑦 ′′ + 𝑃(𝑥 )𝑦 ′ + 𝑄 (𝑥 )𝑦 = 0.
𝐵(𝑥) 𝐶(𝑥)
Here, 𝑃 (𝑥 ) = and 𝑄 (𝑥 ) = , but what happens if there is a point
𝐴(𝑥) 𝐴(𝑥)
where 𝐴(𝑥 ) = 0?
Ex. 𝑥 2 𝑦 ′′ + 𝑦 ′ + 𝑥 3 𝑦 = 0, dividing by 𝑥 2 we get:
1
𝑦 ′′ + 𝑦′ + 𝑥𝑦 = 0.
𝑥2
1
𝑃(𝑥 ) = is not analytic at 𝑥 = 0.
𝑥2
Def. If we consider the equation: 𝑦 ′′ + 𝑃(𝑥)𝑦 ′ + 𝑄(𝑥)𝑦 = 0, then 𝑥 = 𝑎 is
called an ordinary point of this equation (and of the equation
𝐴(𝑥 )𝑦 ′′ + 𝐵(𝑥 )𝑦 ′ + 𝐶 (𝑥 )𝑦 = 0) if 𝑃(𝑥 ) and 𝑄 (𝑥 ) are both analytic at
𝑥 = 𝑎. Otherwise 𝑥 = 𝑎 is called a singular point.
1
Ex. For the equation 𝑦 ′′ + 𝑦 ′ + 𝑥𝑦 = 0, 𝑥 = 0 is a singular point and
𝑥2
𝑥 ≠ 0 are ordinary points.
, 2
Ex. For the equation (𝑥 2 − 1)𝑦 ′′ + (3𝑥 2 + 1)𝑦 ′ + (5 − 𝑥 )𝑦 = 0,
𝑥 = 0 is an ordinary point but 𝑥 = ±1 are singular points because:
′′ (3𝑥2 +1) (5−𝑥)
𝑦 + 𝑦′ + 𝑦=0
(𝑥2 −1) (𝑥2 −1)
𝑃(𝑥) and 𝑄(𝑥) don’t have convergent Taylor series around
𝑥 = ±1, but they do for 𝑥 ≠ ±1 (which includes 𝑥 = 0).
So 𝑥 ≠ ±1 are all ordinary points.
Theorem: Suppose that 𝑎 is an ordinary point of the equation,
𝐴(𝑥 )𝑦 ′′ + 𝐵(𝑥 )𝑦 ′ + 𝐶 (𝑥 )𝑦 = 0.
𝐵(𝑥) 𝐶(𝑥)
That is, 𝑃 (𝑥 ) = and 𝑄 (𝑥 ) = are analytic at 𝑥 = 𝑎.
𝐴(𝑥) 𝐴(𝑥)
Then the differential equation has two linearly independent solutions each
of the form:
𝑦(𝑥 ) = ∑∞ 𝑛
𝑛=0 𝑐𝑛 (𝑥 − 𝑎) .
The radius of convergence is at least as large as the distance of 𝑎 to the
nearest (real or complex) singular point.
Ex. Determine the radius of convergence guaranteed by the previous
theorem of a series solution of: (𝑥 2 + 25)𝑦 ′′ + 𝑥𝑦 ′ + 𝑥 2 𝑦 = 0 in
powers of 𝑥. What if it’s in powers of 𝑥 − 4?
𝑥 𝑥2
𝑃(𝑥 ) = 2 ; 𝑄 (𝑥 ) =
𝑥 +25 𝑥2 +25
𝑃(𝑥) and 𝑄(𝑥) have singular points at 𝑥 = ±5𝑖.
Series Solutions Near Ordinary Points
A general homogeneous second order linear differential equation with analytic
coefficients (i.e. the functions can be represented by power series) has the form:
𝐴(𝑥 )𝑦 ′′ + 𝐵(𝑥 )𝑦 ′ + 𝐶 (𝑥 )𝑦 = 0.
If 𝐴(𝑥 ) ≠ 0 for any 𝑥 in an open interval, 𝐼, then we can divide the equation by
𝐴(𝑥 ) to get:
𝑦 ′′ + 𝑃(𝑥 )𝑦 ′ + 𝑄 (𝑥 )𝑦 = 0.
𝐵(𝑥) 𝐶(𝑥)
Here, 𝑃 (𝑥 ) = and 𝑄 (𝑥 ) = , but what happens if there is a point
𝐴(𝑥) 𝐴(𝑥)
where 𝐴(𝑥 ) = 0?
Ex. 𝑥 2 𝑦 ′′ + 𝑦 ′ + 𝑥 3 𝑦 = 0, dividing by 𝑥 2 we get:
1
𝑦 ′′ + 𝑦′ + 𝑥𝑦 = 0.
𝑥2
1
𝑃(𝑥 ) = is not analytic at 𝑥 = 0.
𝑥2
Def. If we consider the equation: 𝑦 ′′ + 𝑃(𝑥)𝑦 ′ + 𝑄(𝑥)𝑦 = 0, then 𝑥 = 𝑎 is
called an ordinary point of this equation (and of the equation
𝐴(𝑥 )𝑦 ′′ + 𝐵(𝑥 )𝑦 ′ + 𝐶 (𝑥 )𝑦 = 0) if 𝑃(𝑥 ) and 𝑄 (𝑥 ) are both analytic at
𝑥 = 𝑎. Otherwise 𝑥 = 𝑎 is called a singular point.
1
Ex. For the equation 𝑦 ′′ + 𝑦 ′ + 𝑥𝑦 = 0, 𝑥 = 0 is a singular point and
𝑥2
𝑥 ≠ 0 are ordinary points.
, 2
Ex. For the equation (𝑥 2 − 1)𝑦 ′′ + (3𝑥 2 + 1)𝑦 ′ + (5 − 𝑥 )𝑦 = 0,
𝑥 = 0 is an ordinary point but 𝑥 = ±1 are singular points because:
′′ (3𝑥2 +1) (5−𝑥)
𝑦 + 𝑦′ + 𝑦=0
(𝑥2 −1) (𝑥2 −1)
𝑃(𝑥) and 𝑄(𝑥) don’t have convergent Taylor series around
𝑥 = ±1, but they do for 𝑥 ≠ ±1 (which includes 𝑥 = 0).
So 𝑥 ≠ ±1 are all ordinary points.
Theorem: Suppose that 𝑎 is an ordinary point of the equation,
𝐴(𝑥 )𝑦 ′′ + 𝐵(𝑥 )𝑦 ′ + 𝐶 (𝑥 )𝑦 = 0.
𝐵(𝑥) 𝐶(𝑥)
That is, 𝑃 (𝑥 ) = and 𝑄 (𝑥 ) = are analytic at 𝑥 = 𝑎.
𝐴(𝑥) 𝐴(𝑥)
Then the differential equation has two linearly independent solutions each
of the form:
𝑦(𝑥 ) = ∑∞ 𝑛
𝑛=0 𝑐𝑛 (𝑥 − 𝑎) .
The radius of convergence is at least as large as the distance of 𝑎 to the
nearest (real or complex) singular point.
Ex. Determine the radius of convergence guaranteed by the previous
theorem of a series solution of: (𝑥 2 + 25)𝑦 ′′ + 𝑥𝑦 ′ + 𝑥 2 𝑦 = 0 in
powers of 𝑥. What if it’s in powers of 𝑥 − 4?
𝑥 𝑥2
𝑃(𝑥 ) = 2 ; 𝑄 (𝑥 ) =
𝑥 +25 𝑥2 +25
𝑃(𝑥) and 𝑄(𝑥) have singular points at 𝑥 = ±5𝑖.
