Series Solutions of Linear Equations

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• Chapter 6
• Series Solutions of Linear Equations

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Outline

• Using power series to solve a differential
  equation. First, we should decide the point we
  choose to be the expanding point that is ordinary
  or not.
• If the point is not an ordinary point, decide it
  a regular or irregular singular point, then use
  the Frobenius series to solve the problem.
• Introduce the Bessel equation and the Legendres
  equation.

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Introduction

• In applications, higher order linear equations
  with variable coefficients are just as important
  as, if not more important than, differential
  equations with constant coefficient.
• Considering a equation it does
  not possess elementary solutions. But we can find
  two linear independent solutions of
  
• by using the series
  expansion.

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6.1 Solutions About Ordinary Points

• In section 4.7, without understanding that the
  most higher-order ordinary equations with
  variable coefficients cannot be solved in terms
  of elementary functions.
• The usual strategy for solving differential
  equations of this sort is to assume a solution in
  the form of an infinite series and proceed in a
  manner similar to the method of undetermined
  coefficients.

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6.1.1 Review of Power Series

• Definition
• A power series in is an infinite series
  of the form
• Such a series is also said to be a power
  series centered at a.
• For example, the power series
  is centered at a 1.
• Convergence
• A power series is convergent
  at a specified value of x if its
• sequence of partial sums
  converges- that is,
• If the limit does not exist at x, the series is
  said to be divergent.

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• Interval of Convergence
• Every power series has an interval of
  convergence. The interval of convergence is the
  set of all real number x for which the series
  converges.
• Note We will use the ratio test to see the
  series is convergence or divergence for
• Radius of Convergence
• As we mentioned that the R is assigned to be an
  interval boundary to check the series for its
  convergence property and R is also called the
  radius of convergence.
• Whats the meaning for the value R?
• It means a distance from the point x to the
  nearest singular point.(see in Theorem 6.1)
• Bringing a concept, the singular point possess
  between convergent and
• divergent region.

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• If then a power series
  converges for and
• diverges for
• For example, if the series converges for xa or
  for all x, then R is equal to 0 or
• Recall that is equivalent to
  
• Note A power series may or may not converge at
  the endpoints a-R or aR of this
• interval.
• Absolute Convergence
• Within its interval of convergence a power series
  converges absolutely.

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• Ratio test
• Convergence of a power series
  can often be determined by the ratio test.
• Suppose that
• If Llt1 the series converges absolutely, if Lgt1
  the series diverges, and if L1 the test is
  inconclusive.

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• Example
• A power series the ratio
  test gives
• The series converges absolutely for
  (Llt1), we get
• The series diverges for Lgt1, that is
• Test for the convergence of the boundary for x1
  or 9.

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• A Power Series Defines a Function
• A power series defines a function
  whose domain is the
• interval of convergence of the series. If
  the radius of the convergence is
• Rgt0, then f is continuous, differential, and
  integrable on the interval
• Thus, and
  can be found by term-by-term
  differentiation and integration.
• If is a power series in x,
  then the first two derivatives are
• It will be useful to substitute
  into the differential equation.

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• Identity Property
• Analytic at a Point
• A function f is analytic at a point a if it can
  be represented by a power series in x-a with a
  positive or infinite radius of convergence.
• For example,

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• Arithmetic of Power Series
• Power series can be combined through the
  operations of addition, multiplication, and
  division.
• Example

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• Shifting the Summation Index
• It is important to combine two or more summations
  with different index, so it may need to shift the
  summation index. You may see the rule by the
  following example.
• Example 1Adding Two Power Series

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6.1.2 Power Series Solutions

• Definition 6.1 Ordinary and Singular Points
• A point is said to be an ordinary point of
  the differential equation
• if
  both P(x) and Q(x) in the standard form
• are
  analytic at A point that is not an
  ordinary point is said to be a singular point of
  the equation.
• Considering the differential equation
  and

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• Polynomial Coefficients

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• Theorem 6.1 Existence of Power Series Solutions
• If is an ordinary point of the
  differential equation
• 
  we can always find two linearly independent
  solutions in the form of a power series centered
  at -that is,
• A series
  solution converges at least on some interval
• defined by whereas R is
  the distance from to the closest
• singular point.
• A solution of the form
  is said to be a solution about the ordinary point
  

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• Example 2. Power Series Solutions
• Solve

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• Example 3. Power Series Solution
• Solve

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• Example 4. Three-Term Recurrence Relation
• If we seek a power series solution
  for the differential equation

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• Nonpolynomial Coefficients
• The next example illustrates how to find a power
  series solution about the ordinary point
  of a differential equation when its
  coefficients are not polynomials.
• Example 5. ODE with Nonpolynomial Coefficients
• Solve

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• Solution Curves
• The approximate graph of a power series solution
  can be
• obtained in several ways. We can always
  resort to graphing the terms in the sequence of
  partial sums of the seriesin other words, the
  graphs of the polynomials
• For a large value of N,
• By this, will
  give us some information about the
• behavior of y(x) near the ordinary point.

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• Remarks
• Even though we can generate as many terms as
  desired in
• series solution either
  through the use of a
• recurrence relation or, as in Example 4, by
  multiplication, it
• may not be possible to deduce any general
  term for the
• coefficients We may have to settle, as
  we did in Example 4
• and 5, for just writing out the first few
  terms of the series.

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6.2 Solutions About Singular Points

• The two differential equations
  are similar only in that they are
  both examples of simple linear second-order
  equations with variable coefficients.
• We saw in the preceding section that since x0 is
  an ordinary point of the first equation, there is
  no problem in finding two linear independent
  power series solutions centered at that point.
• In the contrast, because x0 is a singular
  point(which is defined in Definition 6.1) of the
  second ODE, finding two infinite series solutions
  of the equation about that point becomes a more
  difficult task.

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• Regular and Irregular Singular Points
• Definition 6.2 Regular and Irregular Singular
  Points

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• Example 1. Classification of Singular Points

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• Theorem 6.2 Frobenius Theorem
• If is a regular singular point of
  the differential equation
• 
  then there exists at least one solution of the
  
• form
• where the number r is a constant to be
  determined. The series will
• converge at least on some interval
• If we consider a differential equation that has a
  regular singular point,
• then we can substitute
  into the DE like the approach
• we did before by using the power series to
  solve with the ordinary point.

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• Example 2. Two series solutions

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Indicial Equation

• Equation is called the indicial
  equation of the previous example, and the value
  are called the indicial
  roots, or exponents, of the singularity x0.
• In general, after substituting
  into the given
• differential equation and simplifying, the
  indicial equation is a quadratic equation in r
  that results from equating the total coefficient
  of the lowest power of x to zero.
• We solve for the two values of r and substitute
  these value
• into a recurrence relation such as
• By Theorem 6.2, there is at least one
  solution of the assumed
• series form that can be found.

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• Example for Indicial Equation

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Three Cases

• Case I
• Case II

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• Case III If then there always exists
  two linearly independent solutions of
  of the form

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• Example 5.
• Find the general solution of

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Remark

• When the difference of indicial roots
  is a positive integer
• it sometimes pays to iterate the
  recurrence relation using the smaller root
  first.
• Since r is the root of a quadratic equation, it
  could be complex. Here we do not concern this
  case.
• If x0 is an irregular singular point, we may not
  be able to find
• any solution of the form

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6.3 Two Special Equations

• The two differential equations
• occur frequently in advanced studies in
  applied mathematics, physics, and engineering.
• They are called Bessels equation and Legendres
  equation, respectively.
• In solving (1) we shall assume whereas
  in (2) we shall consider only the case when n is
  a nonnegative integer.

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Solution of Bessels Equation

• Substituting into the Bessels
  equation,

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• Example 1. General Solution Not an Integer
  

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• Bessel Functions of the Second Kinds

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Following plot will show us the first and second
kind of Bessel function.
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• Example 2. General Solution an Integer
• Parametric Bessel Equation

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• Example 4. Derivation Using the Series Definition

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• Spherical Bessel Functions

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• Example 5. Spherical Bessel Function with

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• Now we bring out an extra concept that is not
  mentioned in textbook, that is, the Modified
  Bessel Equation.

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• Example.
• Solution of Legendres Equation

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• Example from P305. Q19
• Properties of

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• Conclusion
• Here we point out two type of these special
  function, it will be categorized to represent
  with a special solution, that we will find out in
  the mathematics handbook.