The document discusses series solutions for second order linear differential equations near ordinary and regular singular points. It defines an ordinary point as a point where the functions p(x) and q(x) in the normalized form of the differential equation are analytic. Near an ordinary point, there exist two linearly independent power series solutions of the form Σcn(x-a)n that converge within the radii of convergence of p(x) and q(x). It also discusses finding series solutions near a regular singular point x0=0, where the limits of p(x) and q(x) as x approaches 0 exist. An initial guess of a power series solution with exponent r is made, and the

1. Series Solutions at Ordinary
point and Regular Singular
point
ADVANCED ENGINEERING MATHEMATICS

2. Series Solutions at Ordinary Point
• We are considering methods of solving second order linear equations when the
coefficients are functions of the independent variable. We consider the second order
linear homogeneous equation
P(x)
𝑑2 𝑦
𝑑𝑥2 + Q(x)
𝑑𝑦
𝑑𝑥
+ R(x)y = 0 (1)
• since the procedure for the non-homogeneous equation is similar. Many problems in
mathematical physics lead to equations of this form having polynomial coefficients;
examples include the Bessel equation
𝑥2y’’ + xy’ + (𝑥2 – 𝑎2) y = 0
• where a is a constant, and the Legendre equation
(1 – 𝑥2) y’’ – 2xy’ + c(c + 1) y = 0
• where c is a constant.

3. • For this reason, as well as to simplify the algebraic computations, we primarily
consider the case in which P, Q, and R are polynomials. However, we will see that
the method can be applied when P, Q, and R are analytic functions. For the time
being, we assume P, Q, and R are polynomials and they do not have common
factors. Suppose that we wish to solve equation (1) in a neighborhood of a point a.
The solution of equation (1) in an interval containing point a is closely related
with the behavior of P(x) in the interval.
• Definition: Given the equation
P(x)
𝑑2 𝑦
𝑑𝑥2 + Q(x)
𝑑𝑦
𝑑𝑥
+ R(x)y = 0
the equation
𝑑2 𝑦
𝑑𝑥2 +
Q(x)
P(x)
𝑑𝑦
𝑑𝑥
+
R(x)
P(x)
y = 0 or
𝑑2 𝑦
𝑑𝑥2 + p(x)
𝑑𝑦
𝑑𝑥
+ q(x) y = 0
where p(x) =
Q(x)
P(x)
and q(x) =
R(x)
P(x)
is called the equivalent normalized form of equation(1)

4. • Definition:
The point a is called an ordinary point of equation (1) if both of the functions p(x) and
q(x) in the equivalent normalized form, are analytic functions at the point a.
If either or both of these functions are not analytic at a, then the point a is a singular
point of equation (1).
• Theorem:
If a is an ordinary point of the differential equation (1), then p(x) = Q(x)/P(x) and q(x)
= R(x)/P(x) are analytic at a, and there are two nontrivial linearly independent power
series solutions of equation (1) of the form
𝑛=0
∞
𝑐 𝑛(𝑥 − 𝑎) 𝑛, 𝑛=0
∞
𝑑 𝑛(𝑥 − 𝑎) 𝑛
Furthermore, the radius of convergence for each of the series solutions is at least as
large as the minimum of the radii of convergence of the series for p(x) and q(x).

5. The Method of Solution
• We assume that a is an ordinary point of the equation (1), so it has a series solution
near a of the form 𝑛=0
∞
𝑐 𝑛(𝑥 − 𝑎) 𝑛 that converges for |x –a| < ρ .
• We want to determine the coefficients c0, c1, c2, … in the series solution. Let’s
differentiate the series termwise twice.
• y(x) = 𝑛=0
∞
𝑐 𝑛(𝑥 − 𝑎) 𝑛 = 𝑐0 + 𝑐1(x-a) + 𝑐2(x−a)2
+ ....
• y'(x) = 𝑛=0
∞
𝑐 𝑛 𝑛(𝑥 − 𝑎) 𝑛−1 = 𝑐1 + 2𝑐2(x-a) + 3𝑐3(x−a)2
+ ....
• y‘’(x) = 𝑛=0
∞
𝑐 𝑛 𝑛(𝑛 − 1)(𝑥 − 𝑎) 𝑛−2 = 2𝑐2 + 6𝑐3(x-a) + 12𝑐4(x−a)2
+ ....
Substituting in the differential equation (1), we get
𝐾0 + 𝐾1(x-a) + 𝐾2(x−a)2
+ .... + 𝐾 𝑛(x−a) 𝑛
+ .... = 0 (4)
where 𝐾 𝑛 is function of certain coefficients 𝑐𝑖. In order that series (4) be valid for all
x in some interval |x – a| < ρ, we must have
𝐾0= 𝐾1= ... = 𝐾 𝑛 = ... = 0

6. Translated Series Solutions
• If we look for a solution of the equation P(x)
𝑑2 𝑦
𝑑𝑥2 + Q(x)
𝑑𝑦
𝑑𝑥
+ R(x)y = 0 of the form
y = 𝑛=0
∞
𝑐 𝑛(𝑥 − 𝑎) 𝑛
, we can make a change of variable x – a = t, obtaining a new
differential equation for y as a function of t, and then look for solutions of this new
equation of the form y(t) = 𝑛=0
∞
𝑐 𝑛 𝑡 𝑛
. When we have finished the calculations, we
replace t by x –a

7. Series Solutions Near a Regular Singular Point
We will now consider solving the equation
P(x)
𝑑2 𝑦
𝑑𝑥2 + Q(x)
𝑑𝑦
𝑑𝑥
+ R(x)y = 0
Where 𝑥0= 0 is a regular singular point. This means that the limits
𝑝0 = lim
𝑥→0
𝑥
Q(x)
P(x)
……(1) and 𝑞0 = lim
𝑥→0
𝑥2 R(x)
P(x)
……(2)
• exist. Sometimes, it is convenient to rewrite this in the form (by dividing by P(x)
and multiplying by 𝑥2):
𝑥2 𝑑2 𝑦
𝑑𝑥2 + x[xp(x)]
𝑑𝑦
𝑑𝑥
+ [𝑥2
q(x)]y = 0 ……(3)
Our initial guess for a solution to this equation is

8. • y(x) = 𝑛=0
∞
𝑎 𝑛 𝑥 𝑛+𝑟
• Where 𝑎0 ≠ 0 After taking derivatives and plugging y’ and y’’ into equation (3),
we can find the indicial equation F(r)=0 by looking at the coeffcient of 𝑥 𝑟, which
is equal to 𝑎0 𝐹 𝑟 .The indicial equation can also be found directly from the
equation:
𝐹 𝑟 = r(r-1)+ 𝑝0 𝑟 + 𝑝0 = 0
• Where 𝑝0 And 𝑞0 are the limits in equations (1) and (2). The roots of
𝐹 𝑟 , denoted by 𝑟1and 𝑟2, are called the exponents at the singularity.
Suppose 𝑟1 ≥ 𝑟2 and are real. Then in the interval 0< 𝑥 < p ,where p is the
minimum of the radii of convergence, there is a solution of the form:
𝑦1(x) = 𝑥 𝑟1 1 + 𝑛=0
∞
𝑎 𝑛(𝑟1)𝑥 𝑛
• Where 𝑎 𝑛(𝑟1) is obtained from the recurrence relation one gets from
looking at the coefficient of 𝑥 𝑛+𝑟 and plugging in r = r1 And 𝑎0 = 1

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