Differential Equations-Series Solutions Near Regular Singular Points the Frobenius Method, guaranteed and verified 100% Pass

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Series Solutions Near Regular Singular Points: The Frobenius Method


If 𝐴(𝑥 )𝑦 ′′ + 𝐵 (𝑥 )𝑦 ′ + 𝐶 (𝑥 )𝑦 = 0 and 𝐴, 𝐵, and 𝐶 have no common factors
then points where 𝐴(𝑥 ) = 0 are singular points of this equation.



Ex. (1 − 𝑥 2 )𝑦 ′′ − 2𝑥𝑦 ′ + 𝑛(𝑛 + 1)𝑦 = 0 (Legendre’s Equation) has
singular points at 𝑥 = ±1.



We will focus our attention on situations where 𝑥 = 0 is the singular point. If
𝑥 = 𝑎 were a singular point we could always make a substitution, 𝑡 = 𝑥 − 𝑎,
which would have a singular point at 𝑡 = 0.



We will consider equations of the form: 𝐴(𝑥 )𝑦 ′′ + 𝐵 (𝑥 )𝑦 ′ + 𝐶 (𝑥 )𝑦 = 0,
where 𝐴, 𝐵, and 𝐶 are analytic at 𝑥 = 0 (i.e. 𝐴(𝑥), 𝐵(𝑥), and 𝐶(𝑥) have
convergent power series in 𝑥 around 𝑥 = 0).



In general, if 𝐴(𝑥 ) = 0 at 𝑥 = 0, we will not be able to solve the equation with
a power series. However, in certain circumstances we will be able to generalize
the power series approach.

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Ex. Bessel’s Equation has a singularity at 𝑥 = 0.

𝑥 2 𝑦′′ + 𝑥𝑦 ′ + 𝑥 2 𝑦 = 0
or
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𝑦 ′′ + 𝑦 ′ + 𝑦 = 0.
𝑥



If we take 𝐴(𝑥 )𝑦 ′′ + 𝐵 (𝑥 )𝑦 ′ + 𝐶 (𝑥 )𝑦 = 0 and divide by 𝐴(𝑥 ) we get:

𝑦 ′′ + 𝑃(𝑥 )𝑦 ′ + 𝑄 (𝑥 )𝑦 = 0
𝐵(𝑥) 𝐶(𝑥)
where 𝑃 (𝑥 ) = , 𝑄 (𝑥 ) = .
𝐴(𝑥) 𝐴(𝑥)
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In our example, 𝑃 (𝑥 ) = , 𝑄 (𝑥 ) = 1.
𝑥



We will see that we will be able to generalize the power series approach if 𝑃 (𝑥 )
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approaches infinity no more rapidly than and 𝑄 (𝑥 ) approaches infinity no
𝑥
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more rapidly than as 𝑥 goes to zero from the right.
𝑥2
If we rewrite 𝑦 ′′ + 𝑃(𝑥)𝑦 ′ + 𝑄(𝑥)𝑦 = 0 in the form:
𝑝(𝑥 ) ′ 𝑞(𝑥 )
𝑦 ′′ + 𝑦 + 2 𝑦=0
𝑥 𝑥
where 𝑝(𝑥 ) = 𝑥𝑃 (𝑥 ) and 𝑞 (𝑥 ) = 𝑥 2 𝑄 (𝑥 ), then we have the
following definition.



Def. The singular point 𝑥 = 0 is a regular singular point if the functions 𝑝(𝑥 )

and 𝑞 (𝑥 ) are both analytic at 𝑥 = 0. Otherwise, it is an irregular singular
point.