CHAPTER 5

Series Solutions of ODEs. Special Functions

In the previous chapters, we have seen that linear ODEs with constant coefficients can be solved by algebraic methods, and that their solutions are elementary functions known from calculus. For ODEs with variable coefficients the situation is more complicated, and their solutions may be nonelementary functions. Legendre's, Bessel's, and the hypergeometric equations are important ODEs of this kind. Since these ODEs and their solutions, the Legendre polynomials, Bessel functions, and hypergeometric functions, play an important role in engineering modeling, we shall consider the two standard methods for solving such ODEs.

The first method is called the power series method because it gives solutions in the form of a power series a₀ + a₁x + a₂x² + a₃x³ + ….

The second method is called the Frobenius method and generalizes the first; it gives solutions in power series, multiplied by a logarithmic term or a fractional power x^r, in cases such as Bessel's equation, in which the first method is not general enough.

All those more advanced solutions and various other functions not appearing in calculus are known as higher functions or special functions, which has become a technical term. Each of these functions is important enough to give it a name and investigate its properties and relations to other functions in great detail (take a look into Refs. [GenRef1], [GenRef10], or [All] in App. 1). Your CAS knows practically all functions you will ever need in industry or research labs, but it is up to you to find your way through this vast terrain of formulas. The present chapter may give you some help in this task.

Prerequisite: Chap. 2.

Section that may be omitted in a shorter course: 5.5.

References and Answers to Problems: App. 1 Part A, and App. 2.

5.1 Power Series Method

The power series method is the standard method for solving linear ODEs with variable coefficients. It gives solutions in the form of power series. These series can be used for computing values, graphing curves, proving formulas, and exploring properties of solutions, as we shall see. In this section we begin by explaining the idea of the power series method.

From calculus we remember that a power series (in powers of x − x₀) is an infinite series of the form

Here, x is a variable. a₀, a₁, a₂, … are constants, called the coefficients of the series. x₀ is a constant, called the center of the series. In particular, if x₀ = 0, we obtain a power series in powers of x

We shall assume that all variables and constants are real.

We note that the term “power series” usually refers to a series of the form (1) [or (2)] but does not include series of negative or fractional powers of x. We use m as the summation letter, reserving n as a standard notation in the Legendre and Bessel equations for integer values of the parameter.

Idea and Technique of the Power Series Method

The idea of the power series method for solving linear ODEs seems natural, once we know that the most important ODEs in applied mathematics have solutions of this form. We explain the idea by an ODE that can readily be solved otherwise.

We now describe the method in general and justify it after the next example. For a given ODE

we first represent p(x) and q(x) by power series in powers of x (or of x − x₀ if solutions in powers of x − x₀ are wanted). Often p(x) and q(x) are polynomials, and then nothing needs to be done in this first step. Next we assume a solution in the form of a power series (2) with unknown coefficients and insert it as well as (3) and

into the ODE. Then we collect like powers of x and equate the sum of the coefficients of each occurring power of x to zero, starting with the constant terms, then taking the terms containing x, then the terms in x², and so on. This gives equations from which we can determine the unknown coefficients of (3) successively.

Theory of the Power Series Method

The nth partial sum of (1) is

where n = 0, 1, …. If we omit the terms of s_n from (1), the remaining expression is

This expression is called the remainder of (1) after the term a_n(x − x₀)ⁿ.

For example, in the case of the geometric series

we have

In this way we have now associated with (1) the sequence of the partial sums s₀(x), s₁(x), s₂(x), …. If for some x = x₁ this sequence converges, say,

then the series (1) is called convergent at x = x₁, the number s(x₁) is called the value or sum of (1) at x₁, and we write

Then we have for every n,

If that sequence diverges at x = x₁, the series (1) is called divergent at x = x₁.

In the case of convergence, for any positive there is an N (depending on ) such that, by (8)

Geometrically, this means that all s_n(x₁) with n > N lie between s(x₁) − and s(x₁) + (Fig. 104). Practically, this means that in the case of convergence we can approximate the sum s(x₁) of (1) at x₁ by s_x(x₁) as accurately as we please, by taking n large enough.

Fig. 104. Inequality (9)

Where does a power series converge? Now if we choose x = x₀ in (1), the series reduces to the single term a₀ because the other terms are zero. Hence the series converges at. In some cases this may be the only value of x for which (1) converges. If there are other values of x for which the series converges, these values form an interval, the convergence interval. This interval may be finite, as in Fig. 105, with midpoint x₀. Then the series (1) converges for all x in the interior of the interval, that is, for all x for which

and diverges for |x − x₀| > R. The interval may also be infinite, that is, the series may converge for all x.

Fig. 105. Convergence interval (10) of a power series with center x₀

The quantity R in Fig. 105 is called the radius of convergence (because for a complex power series it is the radius of disk of convergence). If the series converges for all x, we set R = ∞ (and 1/R = 0).

The radius of convergence can be determined from the coefficients of the series by means of each of the formulas

provided these limits exist and are not zero. [If these limits are infinite, then (1) converges only at the center x₀.]

When do power series solutions exist? Answer: if p, q, r in the ODEs

have power series representations (Taylor series). More precisely, a function f(x) is called analytic at a point x = x₀ if it can be represented by a power series in powers of x − x₀ with positive radius of convergence. Using this concept, we can state the following basic theorem, in which the ODE (12) is in standard form, that is, it begins with the y″. If your ODE begins with, say, h(x)y″, divide it first by h(x) and then apply the theorem to the resulting new ODE.

The proof of this theorem requires advanced complex analysis and can be found in Ref. [A11] listed in App. 1.

We mention that the radius of convergence R in Theorem 1 is at least equal to the distance from the point x = x₀ to the point (or points) closest to x₀ at which one of the functions p, q, r, as functions of a complex variable, is not analytic. (Note that that point may not lie on the x-axis but somewhere in the complex plane.)

Further Theory: Operations on Power Series

In the power series method we differentiate, add, and multiply power series, and we obtain coefficient recursions (as, for instance, in Example 3) by equating the sum of the coefficients of each occurring power of x to zero. These four operations are permissible in the sense explained in what follows. Proofs can be found in Sec. 15.3.

1. Termwise Differentiation.A power series may be differentiated term by term. More precisely: if

converges for |x − x₀| < R, where R > 0, then the series obtained by differentiating term by term also converges for those x and represents the derivative y′ of y for those x:

Similarly for the second and further derivatives.

2. Termwise Addition.Two power series may be added term by term. More precisely: if the series

have positive radii of convergence and their sums are f(x) and g(x), then the series

converges and represents f(x) + g(x) for each x that lies in the interior of the convergence interval common to each of the two given series.

3. Termwise Multiplication.Two power series may be multiplied term by term. More precisely: Suppose that the series (13) have positive radii of convergence and let f(x) and g(x) be their sums. Then the series obtained by multiplying each term of the first series by each term of the second series and collecting like powers of x − x₀, that is,

converges and represents f(x)g(x) for each x in the interior of the convergence interval of each of the two given series.

4. Vanishing of All Coefficients (“Identity Theorem for Power Series.”) If a power series has a positive radius of convergent convergence and a sum that is identically zero throughout its interval of convergence, then each coefficient of the series must be zero.

5.2 Legendre's Equation. Legendre Polynomials P_n(x)

Legendre's differential equation¹

is one of the most important ODEs in physics. It arises in numerous problems, particularly in boundary value problems for spheres (take a quick look at Example 1 in Sec. 12.10).

The equation involves a parameter n, whose value depends on the physical or engineering problem. So (1) is actually a whole family of ODEs. For n = 1 we solved it in Example 3 of Sec. 5.1 (look back at it). Any solution of (1) is called a Legendre function. The study of these and other “higher” functions not occurring in calculus is called the theory of special functions. Further special functions will occur in the next sections.

Dividing (1) by 1 − x², we obtain the standard form needed in Theorem 1 of Sec. 5.1 and we see that the coefficients −2x/(1 − x²) and n(n + 1)/(1 − x²) of the new equation are analytic at x = 0, so that we may apply the power series method. Substituting

and its derivatives into (1), and denoting the constant n(n + 1) simply by k, we obtain

By writing the first expression as two separate series we have the equation

It may help you to write out the first few terms of each series explicitly, as in Example 3 of Sec. 5.1; or you may continue as follows. To obtain the same general power x^s in all four series, set m − 2 = s (thus m = s + 2) in the first series and simply write s instead of m in the other three series. This gives

(Note that in the first series the summation begins with s = 0.) Since this equation with the right side 0 must be an identity in x if (2) is to be a solution of (1), the sum of the coefficients of each power of x on the left must be zero. Now x⁰ occurs in the first and fourth series only, and gives [remember that k = n(n + 1)]

x¹ occurs in the first, third, and fourth series and gives

The higher powers x², x³, … occur in all four series and give

The expression in the brackets […] can be written (n − s)(n + s + 1), as you may readily verify. Solving (3a) for a₂ and (3b) for a₃ as well as (3c) for a_s+2, we obtain the general formula

This is called a recurrence relation or recursion formula. (Its derivation you may verify with your CAS.) It gives each coefficient in terms of the second one preceding it, except for a₀ and a₁, which are left as arbitrary constants. We find successively

and so on. By inserting these expressions for the coefficients into (2) we obtain

where

These series converge for |x| < 1 (see Prob. 4; or they may terminate, see below). Since (6) contains even powers of x only, while (7) contains odd powers of x only, the ratio y₁/y₂ is not a constant, so that y₁ and y₂ are not proportional and are thus linearly independent solutions. Hence (5) is a general solution of (1) on the interval −1 < x < 1.

Note that x = ±1 are the points at which 1 − x² = 0, so that the coefficients of the standardized ODE are no longer analytic. So it should not surprise you that we do not get a longer convergence interval of (6) and (7), unless these series terminate after finitely many powers. In that case, the series become polynomials.

Polynomial Solutions. Legendre Polynomials P_n(x)

The reduction of power series to polynomials is a great advantage because then we have solutions for all x, without convergence restrictions. For special functions arising as solutions of ODEs this happens quite frequently, leading to various important families of polynomials; see Refs. [GenRef1], [GenRef10] in App. 1. For Legendre's equation this happens when the parameter n is a nonnegative integer because then the right side of (4) is zero for s = n, so that a_n+4 = 0, a_n+6 = 0, …. Hence if n is even, y₁(x) reduces to a polynomial of degree n. If n is odd, the same is true for y₂(x). These polynomials, multiplied by some constants, are called Legendre polynomials and are denoted by P_n(x). The standard choice of such constants is done as follows. We choose the coefficient a_n of the highest power xⁿ as

(and a_n = 1 if n = 0). Then we calculate the other coefficients from (4), solved for a_s in terms of a_s+2, that is,

The choice (8) makes P_n(1) = 1 for every n (see Fig. 107); this motivates (8). From (9) with s = n − 2 and (8) we obtain

Using (2n)! = 2n(2n − 1)(2n − 2)! in the numerator and n! = n(n − 1)! and n! = n(n − 1)(n − 2)! in the denominator, we obtain

n(n − 1)2n(2n − 1) cancels, so that we get

Similarly,

and so on, and in general, when n − 2m 0,

The resulting solution of Legendre's differential equation (1) is called the Legendre polynomialof degree n and is denoted by P_n(x).

From (10) we obtain

where M = n/2 or (n − 1)/2, whichever is an integer. The first few of these functions are (Fig. 107)

and so on. You may now program (11) on your CAS and calculate P_n(x) as needed.

Fig. 107. Legendre polynomials

The Legendre polynomials P_n(x) are orthogonal on the interval −1 x 1, a basic property to be defined and used in making up “Fourier–Legendre series” in the chapter on Fourier series (see Secs. 11.5–11.6).

5.3 Extended Power Series Method: Frobenius Method

Several second-order ODEs of considerable practical importance—the famous Bessel equation among them—have coefficients that are not analytic (definition in Sec. 5.1), but are “not too bad,” so that these ODEs can still be solved by series (power series times a logarithm or times a fractional power of x, etc.). Indeed, the following theorem permits an extension of the power series method. The new method is called the Frobenius method.⁴ Both methods, that is, the power series method and the Frobenius method, have gained in significance due to the use of software in actual calculations.

For example, Bessel's equation (to be discussed in the next section)

is of the form (1) with b(x) = 1 and c(x) = x² − v² analytic at x = 0, so that the theorem applies. This ODE could not be handled in full generality by the power series method.

Similarly, the so-called hypergeometric differential equation (see Problem Set 5.3) also requires the Frobenius method.

The point is that in (2) we have a power series times a single power of x whose exponent r is not restricted to be a nonnegative integer. (The latter restriction would make the whole expression a power series, by definition; see Sec. 5.1.)

The proof of the theorem requires advanced methods of complex analysis and can be found in Ref. [A11] listed in App. 1.

Regular and Singular Points. The following terms are practical and commonly used. A regular point of the ODE

is a point x₀ at which the coefficients p and q are analytic. Similarly, a regular point of the ODE

is an x₀ at which are analytic and (so what we can divide by and get the previous standard form). Then the power series method can be applied. If x₀ is not a regular point, it is called a singular point.

Indicial Equation, Indicating the Form of Solutions

We shall now explain the Frobenius method for solving (1). Multiplication of (1) by x² gives the more convenient form

We first expand b(x) and c(x) in power series,

or we do nothing if b(x) and c(x) are polynomials. Then we differentiate (2) term by term, finding

By inserting all these series into (1′) we obtain

We now equate the sum of the coefficients of each power x^r, x^r+1, x^r+2, … to zero. This yields a system of equations involving the unknown coefficients a_m. The smallest power is x^r and the corresponding equation is

Since by assumption a₀ ≠ 0, the expression in the brackets […] must be zero. This gives

This important quadratic equation is called the indicial equation of the ODE (1). Its role is as follows.

The Frobenius method yields a basis of solutions. One of the two solutions will always be of the form (2), where r is a root of (4). The other solution will be of a form indicated by the indicial equation. There are three cases:

Case 1. Distinct roots not differing by an integer 1, 2, 3, ….

Case 2. A double root.

Case 3. Roots differing by an integer 1, 2, 3, ….

Cases 1 and 2 are not unexpected because of the Euler–Cauchy equation (Sec. 2.5), the simplest ODE of the form (1). Case 1 includes complex conjugate roots r₁ and because Im r₁ is imaginary, so it cannot be a real integer. The form of a basis will be given in Theorem 2 (which is proved in App. 4), without a general theory of convergence, but convergence of the occurring series can be tested in each individual case as usual. Note that in Case 2 we must have a logarithm, whereas in Case 3 we may or may not.

Typical Applications

Technically, the Frobenius method is similar to the power series method, once the roots of the indicial equation have been determined. However, (5)–(10) merely indicate the general form of a basis, and a second solution can often be obtained more rapidly by reduction of order (Sec. 2.1).

Fig. 109. Solutions in Example 2

The Frobenius method solves the hypergeometric equation, whose solutions include many known functions as special cases (see the problem set). In the next section we use the method for solving Bessel's equation.

5.4 Bessel's Equation. Bessel Functions J_ν(x)

One of the most important ODEs in applied mathematics in Bessel's equation,⁶

where the parameter ν (nu) is a given real number which is positive or zero. Bessel's equation often appears if a problem shows cylindrical symmetry, for example, as the membranes in Sec. 12.9. The equation satisfies the assumptions of Theorem 1. To see this, divide (1) by x² to get the standard form y″ + y′/x + (1 − ν²/x²)y = 0. Hence, according to the Frobenius theory, it has a solution of the form

Substituting (2) and its first and second derivatives into Bessel's equation, we obtain

We equate the sum of the coefficients of x^s+r to zero. Note that this power x^s+r corresponds to m = s in the first, second, and fourth series, and to m = s − 2 in the third series. Hence for s = 0 and s = 1, the third series does not contribute since m 0.

For s = 2, 3, … all four series contribute, so that we get a general formula for all these s. We find

From (3a) we obtain the indicial equation by dropping a₀,

The roots are r₁ = ν( 0) and r₂ = −ν.

Coefficient Recursion for r = r₁ = ν. For r = ν, Eq. (3b) reduces to (2ν + 1)a₁ = 0. Hence a₁ = 0 since ν 0. Substituting r = ν in (3c) and combining the three terms containing a_s gives simply

Since a₁ = 0 and ν 0, it follows from (5) that a₃ = 0, a₅ = 0, …. Hence we have to deal only with even-numbered coefficients a_s with s = 2m. For s = 2m, Eq. (5) becomes

Solving for a_2m gives the recursion formula

From (6) we can now determine a₂, a₄, … successively. This gives

and so on, and in general

Bessel Functions J_n(x) for Integer ν = n

Integer values of ν are denoted by n. This is standard. For ν = n the relation (7) becomes

a₀ is still arbitrary, so that the series (2) with these coefficients would contain this arbitrary factor a₀. This would be a highly impractical situation for developing formulas or computing values of this new function. Accordingly, we have to make a choice. The choice a₀ = 1 would be possible. A simpler series (2) could be obtained if we could absorb the growing product (n + 1)(n + 2) … (n + m) into a factorial function (n + m)! What should be our choice? Our choice should be

because then n! (n + 1) … (n + m) = (n + m)! in (8), so that (8) simply becomes

By inserting these coefficients into (2) and remembering that c₁ = 0, c₃ = 0, … we obtain a particular solution of Bessel's equation that is denoted by J_n(x):

J_n(x) is called the Bessel function of the first kind of order n. The series (11) converges for all x, as the ratio test shows. Hence J_n(x) is defined for all x. The series converges very rapidly because of the factorials in the denominator.

Bessel Functions J_ν(x) for any ν 0. Gamma Function

We now proceed from integer ν = n to any ν 0. We had a₀ = 1/(2ⁿn!) in (9). So we have to extend the factorial function to any ν 0. For this we choose

with the gamma function Γ(ν + 1) defined by

(CAUTION! Note the convention ν + 1 on the left but ν in the integral.) Integration by parts gives

This is the basic functional relation of the gamma function

Now from (16) with ν = 0 and then by (17) we obtain

and then Γ(2) = 1 · Γ(1) = 1!, Γ(3) = 2Γ(1) = 2! and in general

Hence the gamma function generalizes the factorial function to arbitrary positive ν. Thus (15) with ν = n agrees with (9).

Furthermore, from (7) with a₀ given by (15) we first have

Now (17) gives (ν + 1)Γ(ν + 1) = Γ(ν + 2), (ν + 2)Γ(ν + 2) = Γ(ν + 3) and so on, so that

Hence because of our (standard!) choice (15) of a₀ the coefficients (7) are simply

With these coefficients and r = r₁ = ν we get from (2) a particular solution of (1), denoted by J_ν(x) and given by

J_ν(x) is called the Bessel function of the first kind of order ν. The series (20) converges for all x, as one can verify by the ratio test.

Discovery of Properties from Series

Bessel functions are a model case for showing how to discover properties and relations of functions from series by which they are defined. Bessel functions satisfy an incredibly large number of relationships—look at Ref. [A13] in App. 1; also, find out what your CAS knows. In Theorem 3 we shall discuss four formulas that are backbones in applications and theory.

Bessel Functions J_ν with Half-Integer ν Are Elementary

We discover this remarkable fact as another property obtained from the series (20) and confirm it in the problem set by using Bessel's ODE.

General Solution. Linear Dependence

For a general solution of Bessel's equation (1) in addition to J_ν we need a second linearly independent solution. For ν not an integer this is easy. Replacing ν by −ν in (20), we have

Since Bessel's equation involves ν², the functions J_ν and J_−ν are solutions of the equation for the same ν. If ν is not an integer, they are linearly independent, because the first terms in (20) and in (24) are finite nonzero multiples of x^ν and x^−ν. Thus, if ν is not an integer, a general solution of Bessel's equation for all x ≠ 0 is

This cannot be the general solution for an integer ν = n because, in that case, we have linear dependence. It can be seen that the first terms in (20) and (24) are finite nonzero multiples of x^ν and x^−ν, respectively. This means that, for any integer ν = n, we have linear dependence because

The difficulty caused by (25) will be overcome in the next section by introducing further Bessel functions, called of the second kind and denoted by Y_ν.

5.5 Bessel Functions Y_ν(x). General Solution

To obtain a general solution of Bessel's equation (1), Sec. 5.4, for any, we now introduce Bessel functions of the second kind Y_ν(x), beginning with the case ν = n = 0.

When n = 0, Bessel's equation can be written (divide by x)

Then the indicial equation (4) in Sec. 5.4 has a double root r = 0. This is Case 2 in Sec. 5.3. In this case we first have only one solution, J₀(x). From (8) in Sec. 5.3 we see that the desired second solution must be of the form

We substitute y₂ and its derivatives

into (1). Then the sum of the three logarithmic terms xJ″₀ ln x, J′₀ ln x, and xJ₀ ln x is zero because J₀ is a solution of (1). The terms −J₀/x and J₀/x (from xy″ and y′) cancel. Hence we are left with

Addition of the first and second series gives Σm²A_mx^m−1. The power series of J′₀(x) is obtained from (12) in Sec. 5.4 and the use of m!/m = (m − 1)! in the form

Together with Σm²A_mx^m−1 and ΣA_mx^m+1 this gives

First, we show that the A_m with odd subscripts are all zero. The power x⁰ occurs only in the second series, with coefficient A₁. Hence A₁ = 0. Next, we consider the even powers x^2s. The first series contains none. In the second series, m − 1 = 2s gives the term (2s + 1)²A_2s+1x^2s. In the third series, m + 1 = 2s. Hence by equating the sum of the coefficients of x^2s to zero we have

Since A₁ = 0, we thus obtain A₃ = 0, A₅ = 0, …, successively.

We now equate the sum of the coefficients of x^2s+1 to zero. For s = 0 this gives

For the other values of s we have in the first series in (3*) 2m − 1 = 2s + 1, hence m = s + 1, in the second m − 1 = 2s + 1, and in the third m + 1 = 2s + 1. We thus obtain

For s = 1 this yields

and in general

Using the short notations

and inserting (4) and A₁ = A₃ = … = 0 into (2), we obtain the result

Since J₀ and y₂ are linearly independent functions, they form a basis of (1) for x > 0. Of course, another basis is obtained if we replace y₂ by an independent particular solution of the form a(y₂ + bJ₀), where a (≠ 0) and b are constants. It is customary to choose a = 2/π and b = γ − ln 2, where the number γ = 0.57721566490 … is the so-called Euler constant, which is defined as the limit of

as s approaches infinity. The standard particular solution thus obtained is called the Bessel function of the second kind of order zero (Fig. 112) or Neumann's function of order zero and is denoted by Y₀(x). Thus [see (4)]

For small x > 0 the function Y₀(x) behaves about like ln x (see Fig. 112, why?), and Y₀(x) → −∞ as x → 0.

Bessel Functions of the Second Kind Y_n(x)

For ν = n = 1, 2, … a second solution can be obtained by manipulations similar to those for n = 0, starting from (10), Sec. 5.4. It turns out that in these cases the solution also contains a logarithmic term.

The situation is not yet completely satisfactory, because the second solution is defined differently, depending on whether the order is an integer or not. To provide uniformity of formalism, it is desirable to adopt a form of the second solution that is valid for all values of the order. For this reason we introduce a standard second solution Y_ν(x) defined for all ν by the formula

This function is called the Bessel function of the second kind of order ν or Neumann's function⁷ of order. Figure 112 shows Y₀(x) and Y₁(x).

Let us show that J_ν and Y_ν are indeed linearly independent for all ν (and x > 0).

For noninteger order ν, the function Y_ν(x) is evidently a solution of Bessel's equation because J_ν(x) and J_−ν(x) are solutions of that equation. Since for those ν the solutions J_ν and J_−ν are linearly independent and Y_ν involves J_−ν, the functions J_ν and Y_ν are linearly independent. Furthermore, it can be shown that the limit in (7b) exists and Y_n is a solution of Bessel's equation for integer order; see Ref. [A13] in App. 1. We shall see that the series development of Y_n(x) contains a logarithmic term. Hence J_n(x) and Y_n(x) are linearly independent solutions of Bessel's equation. The series development of Y_n(x) can be obtained if we insert the series (20) in Sec. 5.4 and (2) in this section for J_ν(x) and J_−ν(x) into (7a) and then let ν approach n; for details see Ref. [A13]. The result is

Fig. 112. Bessel functions of the second kind Y₀ and Y₁. (For a small table, see App. 5.)

where x > 0, n = 0, 1, …, and [as in (4)] h₀ = 0, h₁ = 1,

For n = 0 the last sum in (8) is to be replaced by 0 [giving agreement with (6)].

Furthermore, it can be shown that

Our main result may now be formulated as follows.

We finally mention that there is a practical need for solutions of Bessel's equation that are complex for real values of x. For this purpose the solutions

are frequently used. These linearly independent functions are called Bessel functions of the third kind of order ν or first and secondHankel functions⁸ of order ν.

This finishes our discussion on Bessel functions, except for their “orthogonality,” which we explain in Sec. 11.6. Applications to vibrations follow in Sec. 12.10.

CHAPTER 5 REVIEW QUESTIONS AND PROBLEMS

1. Why are we looking for power series solutions of ODEs?
2. What is the difference between the two methods in this chapter? Why do we need two methods?
3. What is the indicial equation? Why is it needed?
4. List the three cases of the Frobenius method, and give examples of your own.
5. Write down the most important ODEs in this chapter from memory.
6. Can a power series solution reduce to a polynomial? When? Why is this important?
7. What is the hypergeometric equation? Where does the name come from?
8. List some properties of the Legendre polynomials.
9. Why did we introduce two kinds of Bessel functions?
10. Can a Bessel function reduce to an elementary function? When?

11–20 POWER SERIES METHOD OR FROBENIUS METHOD

Find a basis of solutions. Try to identify the series as expansions of known functions. Show the details of your work.

• 11. y″ + 4y = 0
• 12. xy″ + (1 − 2x)y′ + (x − 1)y = 0
• 13. (x − 1)²y″ − (x − 1)y′ − 35y = 0
• 14. 16(x + 1)²y″ + 3y = 0
• 15. x²y″ + xy′ + (x² − 5)y = 0
• 16. x²y″ + 2x³y′ + (x² − 2)y = 0
• 17. xy″ − (x + 1)y′ + y = 0
• 18. xy″ + 3y′ + 4x³y = 0
• 19.
• 20. xy″ + y′ − xy = 0

SUMMARY OF CHAPTER 5 Series Solution of ODEs. Special Functions