Chapter 6 Series Solutions of Linear Equations

1. Chapter 6 Series Solutions of Linear Equations 1

2. 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 Legendre’s equation. 2

3. 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. 3

4. 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. 4

5. 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. 5

6. 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. • What’s 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. 6

7. If then a power series converges for and diverges for • For example, if the series converges for x=a 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 a+R of this interval. • Absolute Convergence • Within its interval of convergence a power series converges absolutely. 7

8. Ratio test • Convergence of a power series can often be determined by the ratio test. • Suppose that • If L<1 the series converges absolutely, if L>1 the series diverges, and if L=1 the test is inconclusive. 8

9. Example: A power series the ratio test gives • The series converges absolutely for (L<1), we get • The series diverges for L>1, that is • Test for the convergence of the boundary for x=1 or 9. 9

10. 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 R>0, 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. 10

11. 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, 11

12. Arithmetic of Power Series • Power series can be combined through the operations of addition, multiplication, and division. • Example: 12

13. 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 1:Adding Two Power Series 13

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15. 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 15

16. Polynomial Coefficients 16

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18. 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 18

19. Example 2. Power Series Solutions • Solve 19

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22. Example 3. Power Series Solution • Solve 22

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25. Example 4. Three-Term Recurrence Relation • If we seek a power series solution for the differential equation 25

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27. 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 27

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29. 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 series—in 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. 29

30. 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. 30

31. 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 x=0 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 x=0 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. 31

32. Regular and Irregular Singular Points • Definition 6.2 Regular and Irregular Singular Points 32

33. Example 1. Classification of Singular Points 33

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35. 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. 35

36. Example 2. Two series solutions 36

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39. 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 x=0. • 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. 39

40. Example for Indicial Equation 40

41. Three Cases • Case I: • Case II: 41

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43. Case III: If then there always exists two linearly independent solutions of of the form 43

44. Example 5. • Find the general solution of 44

45. 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 x=0 is an irregular singular point, we may not be able to find any solution of the form 45

46. 6.3 Two Special Equations • The two differential equations occur frequently in advanced studies in applied mathematics, physics, and engineering. • They are called Bessel’s equation and Legendre’s equation, respectively. • In solving (1) we shall assume whereas in (2) we shall consider only the case when n is a nonnegative integer. 46

47. Solution of Bessel’s Equation • Substituting into the Bessel’s equation, 47

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50. Example 1. General Solution: Not an Integer 50