The Solution

Because x = 0 is a regular singular point of Bessel’s equation, we know that there exists at least one solution of the form . Substituting the last expression into (1) then gives

(3)

From (3) we see that the indicial equation is r² − ν² = 0 so that the indicial roots are r₁ = ν and r₂ = −ν. When r₁ = ν, (3) becomes

Therefore, by the usual argument we can write (1 + 2ν)c₁ = 0 and

(k + 2)(k + 2 + 2ν)c_{k + 2} + c_k = 0

or c_k+2 = ,     k = 0, 1, 2, . . . . (4)

The choice c₁ = 0 in (4) implies c₃ = c₅ = c₇ = … = 0, so for k = 0, 2, 4, … we find, after letting k + 2 = 2n, n = 1, 2, 3, . . ., that

. (5)

Thus (6)

It is standard practice to choose c₀ to be a specific value—namely,

c₀ =

where Γ(1 + ν) is the gamma function. See Appendix A. Since this latter function possesses the convenient property Γ(1 + α) = αΓ(α), we can reduce the indicated product in the denominator of (6) to one term. For example,

Hence we can write (6) as

for n = 0, 1, 2, . . . .

Bessel Functions of the First Kind

The series solution is usually denoted by J_ν(x):

(7)

If ν ≥ 0, the series converges at least on the interval [0, ∞). Also, for the second exponent r₂ = −ν we obtain, in exactly the same manner,

(8)

The functions J_ν(x) and J_−ν(x) are called Bessel functions of the first kind of order ν and −ν, respectively. Depending on the value of ν, (8) may contain negative powers of x and hence converge on the interval (0, ∞).^*

Now some care must be taken in writing the general solution of (1). When ν = 0, it is apparent that (7) and (8) are the same. If ν > 0 and r₁ − r₂ = ν − (−ν) = 2ν is not a positive integer, it follows from Case I of Section 5.2 that J_ν(x) and J_−ν(x) are linearly independent solutions of (1) on (0, ∞), and so the general solution on the interval is y = c₁J_ν(x) + c₂J_−ν(x). But we also know from Case II of Section 5.2 that when r₁ − r₂ = 2ν is a positive integer, a second series solution of (1) may exist. In this second case we distinguish two possibilities. When ν = m = positive integer, J_−m(x) defined by (8) and J_m(x) are not linearly independent solutions. It can be shown that J_−m is a constant multiple of J_m (see Property (i) on page 295). In addition, r₁ − r₂ = 2ν can be a positive integer when ν is half an odd positive integer. It can be shown in this latter event that J_ν(x) and J_−ν(x) are linearly independent. In other words, the general solution of (1) on (0, ∞) is

(9)

The graphs of y = J₀(x) (blue) and y = J₁(x) (red) are given in FIGURE 5.3.1.

EXAMPLE 1 General Solution: ν Not an Integer

Bessel Functions of the Second Kind

If ν ≠ integer, the function defined by the linear combination

(10)

and the function J_ν(x) are linearly independent solutions of (1). Thus another form of the general solution of (1) is y = c₁J_ν(x) + c₂Y_ν(x), provided ν ≠ integer. As ν → m, m an integer, (10) has the indeterminate form 0/0. However, it can be shown by L’Hôpital’s rule that lim_ν→m Y_ν(x) exists. Moreover, the function

and J_m(x) are linearly independent solutions of x²y″ + xy′ + (x² − m²)y = 0. Hence for any value of ν the general solution of (1) on the interval (0, ∞) can be written as

(11)

Y_ν(x) is called the Bessel function of the second kind of order ν. FIGURE 5.3.2 shows the graphs of Y₀(x) (blue) and Y₁(x) (red). Because the Bessel function of the second kind of order ν was first introduced in 1867 by the German mathematician Carl Gottfried Neumann (1832–1925), it is also referred to as the Neumann function of order ν. The Neumann function is often denoted by N_ν

EXAMPLE 2 General Solution: ν an Integer

DEs Solvable in Terms of Bessel Functions

Sometimes it is possible to transform a differential equation into equation (1) by means of a change of variable. We can then express the solution of the original equation in terms of Bessel functions. For example, if we let t = αx, α > 0, in

(12)

then by the Chain Rule,

Accordingly (12) becomes

The last equation is Bessel’s equation of order ν with solution y = c₁J_ν(t) + c₂Y_ν(t). By resubstituting t = αx in the last expression we find that the general solution of (12) on the interval (0, ∞) is

(13)

Equation (12), called the parametric Bessel equation of order ν, and its general solution (13) are very important in the study of certain boundary-value problems involving partial differential equations that are expressed in cylindrical coordinates.

Modified Bessel Functions

Another equation that bears a resemblance to (1) is the modified Bessel equation of order ν,

(14)

This DE can be solved in the manner just illustrated for (12). This time if we let t = ix, where i² = −1, then (14) becomes

Since solutions of the last DE are J_ν(t) and Y_ν(t), complex-valued solutions of equation (14) are J_ν(ix) and Y_ν(ix). A real-valued solution, called the modified Bessel function of the first kind of order ν, is defined in terms of J_ν(ix):

See Problem 21 in Exercises 5.3. The general solution of (14) is

(15)

When ν is an integer n the functions I_n(x) and I_−n (x) are not linearly independent on the interval (0, ∞). So, analogous to (10), we define the modified Bessel function of the second kind of order ν ≠ integer to be

(16)

and for integral ν = n,

Because I_ν and K_ν are linearly independent on the interval (0, ∞) for any value of ν, the general solution of (14) is

(17)

The graphs of I₀(x) (blue) and I₁(x) (red) are given in FIGURE 5.3.3 and the graphs K₀(x) (blue) and K₁(x) (red) are shown in FIGURE 5.3.4. Unlike the Bessel functions of the first and second kinds, the graphs of the modified Bessel functions of the first kind and second kind are not oscillatory. Moreover, the graphs in Figures 5.3.3 and 5.3.4 illustrate the fact that the modified Bessel functions I_n(x) and K_n(x), n = 0, 1, 2, . . . have no real zeros in the interval (0, ∞). Also, note that K_n(x) → ∞ as x → 0⁺.

Proceeding as we did in (12) and (13), we see that the general solution of the parametric form of the modified Bessel equation of order ν

(18)

on the interval (0, ∞) is

(19)

EXAMPLE 3 Parametric Modified Bessel Equation

Yet another equation, important because many differential equations fit into its form by appropriate choices of the parameters, is

(20)

Although we shall not supply the details, the general solution of (20),

(21)

can be found by means of a change in both the independent and the dependent variables: z = bx^c, y(x) = w(z). If p is not an integer, then Y_p in (21) can be replaced by J_−p.

EXAMPLE 4 Using (20)

Find the general solution of xy″ + 3y′ + 9y = 0 on (0, ∞).

SOLUTION

By writing the given DE as

we can make the following identifications with (20):

The first and third equations imply a = −1 and c = . With these values the second and fourth equations are satisfied by taking b = 6 and p = 2. From (21) we find that the general solution of the given DE on the interval (0, ∞) is ≡

Aging Spring and Bessel Functions

In Section 3.8 we saw that a mathematical model for the free undamped motion of a mass m on an aging spring is given by We are now in a position to find the general solution of this differential equation.

EXAMPLE 5 Aging Spring Revisited

The other model discussed in Section 5.1 of a spring whose characteristics change with time was mx″ + ktx = 0. By dividing through by m we see that the equation x″ + (k/m) tx = 0 is Airy’s equation, y″ + α²xy = 0. See the discussion following Example 2 in Section 5.1. The general solution of Airy’s differential equation can also be written in terms of Bessel functions. See Problems 34, 35, and 46 in Exercises 5.3.

Properties

We list below a few of the more useful properties of Bessel functions of the first and second kinds of order m, m = 0, 1, 2, . . . :

Note that Property (ii) indicates that J_m(x) is an even function if m is an even integer and an odd function if m is an odd integer. The graphs of Y₀(x) and Y₁(x) in Figure 5.3.2 illustrate Property (iv): Y_m(x) is unbounded at the origin. This last fact is not obvious from (10). The solutions of the Bessel equation of order 0 can be obtained using the solutions y₁(x) in (21) and y₂(x) in (22) of Section 5.2.It can be shown that (21) of Section 5.2 is y₁(x) = J₀(x), whereas (22) of that section is

The Bessel function of the second kind of order 0, Y₀(x), is then defined to be the linear combination for x > 0. That is,

where γ = 0.57721566 . . . is Euler’s constant. Because of the presence of the logarithmic term, it is apparent that Y₀(x) is discontinuous at x = 0.

Numerical Values

The first five nonnegative zeros of J₀(x), J₁(x), Y₀(x), and Y₁(x) are given in Table 5.3.1. Some additional functional values of these four functions are given in Table 5.3.2.

Differential Recurrence Relation

Recurrence formulas that relate Bessel functions of different orders are important in theory and in applications. In the next example we derive a differential recurrence relation.

EXAMPLE 6 Derivation Using the Series Definition

The result in Example 6 can be written in an alternative form. Dividing by x gives

This last expression is recognized as a linear first-order differential equation in J_ν(x). Multiplying both sides of the equality by the integrating factor x^−ν then yields

(22)

It can be shown in a similar manner that

(23)

See Problem 27 in Exercises 5.3. The differential recurrence relations (22) and (23) are also valid for the Bessel function of the second kind Y_ν(x). Observe that when ν = 0 it follows from (22) that

(24)

Results similar to (24) also hold for the modified Bessel functions of the first and second kind of order ν = 0:

(25)

See Problems 43 and 45 in Exercises 5.3 for applications of these derivatives.

Bessel Functions of Half-Integral Order

When the order ν is half an odd integer, that is, . . ., Bessel functions of the first and second kinds can be expressed in terms of the elementary functions sin x, cos x, and powers of x. To see this let’s consider the case when From (7) we have

In view of the properties the gamma function, and the fact that the values of Γ(1 + + n) for n = 0, n = 1, n = 2, and n = 3 are, respectively,

In general,

Hence,

The infinite series in the last line is the Maclaurin series for sin x, and so we have shown that

(26)

We leave it as an exercise to show that

(27)

See Problems 31 and 32 in Exercises 5.3.

If n is an integer, then the order is half an odd integer. Because and , we see from (10) that

(28)

For n = 0 and n = −1 in the last formula, we get, in turn, Y_1/2(x) = −J_−1/2 (x) and Y_−1/2(x) = J_1/2 (x). In view of (26) and (27) these results are the same as

(29)

and (30)

Spherical Bessel Functions

Bessel functions of half-integral order are used to define two more important functions:

     and      (31)

The function j_n(x) is called the spherical Bessel function of the first kind and y_n(x) is the spherical Bessel function of the second kind. For example, by using (26) and (29) we see that for the expressions in (31) become

and            

The graphs of J_n(x) and y_y(x) for n ≥ 0 are very similar to those given in Figures 5.3.1 and 5.3.2; that is, both functions are oscillatory, and y_n(x) becomes unbounded as The graphs of J₀(x) (blue) and y₀(x) (red) are given in FIGURE 5.3.5. See Problems 39 and 40 in Exercises 5.3.

Spherical Bessel functions arise in the solution of a special partial differential equation expressed in spherical coordinates. See Problems 41 and 42 in Exercises 5.3 and Problem 14 in Exercises 14.3.

The Solution

Since x = 0 is an ordinary point of Legendre’s equation (2), we substitute the series , shift summation indices, and combine series to get

which implies that n(n + 1)c₀ + 2c₂ = 0

(n − 1)(n + 2)c₁ + 6c₃ = 0

(j + 2)(j + 1)c_j+2 + (n − j)(n + j + 1)c_j = 0

or (32)

Letting j take on the values 2, 3, 4, . . ., recurrence relation (32) yields

c₄ = −

and so on. Thus for at least |x| < 1 we obtain two linearly independent power series solutions:

(33)

Notice that if n is an even integer, the first series terminates, whereas y₂(x) is an infinite series. For example, if n = 4, then

Similarly, when n is an odd integer, the series for y₂(x) terminates with xⁿ; that is, when n is a nonnegative integer, we obtain an nth-degree polynomial solution of Legendre’s equation.

Since we know that a constant multiple of a solution of Legendre’s equation is also a solution, it is traditional to choose specific values for c₀ or c₁, depending on whether n is an even or odd positive integer, respectively. For n = 0 we choose c₀ = 1, and for n = 2, 4, 6, . . .

c₁ = (−1)^n/2 ;

whereas for n = 1 we choose c₀ = 1, and for n = 3, 5, 7, …,

c₁ = (−1)^(n−1)/2 .

For example, when n = 4 we have

y₁(x) = (−1)^4/2 (35x⁴ − 30x² + 3).

Legendre Polynomials

These specific nth-degree polynomial solutions are called Legendre polynomials and are denoted by P_n(x). From the series for y₁(x) and y₂(x) and from the above choices of c₀ and c₁ we find that the first six Legendre polynomials are

(34)

Remember, P₀(x), P₁(x), P₂(x), P₃(x), …, are, in turn, particular solutions of the differential equations

(35)

n = 3: (1 − x²)y″ − 2xy′ + 12y = 0

⁝ ⁝

The graphs, on the interval [−1, 1], of the six Legendre polynomials in (34) are given in FIGURE 5.3.6.

Properties

You are encouraged to verify the following properties for the Legendre polynomials in (34):

(35)

Property (i) indicates, as is apparent in Figure 5.3.6, that P_n(x) is an even or odd function according to whether n is even or odd.

Recurrence Relation

Recurrence relations that relate Legendre polynomials of different degrees are also important in some aspects of their applications. We state, without proof, the following three-term recurrence relation

(36)

which is valid for k = 1, 2, 3, … . In (34) we listed the first six Legendre polynomials. If, say, we wish to find P₆(x), we can use (36) with k = 5. This relation expresses P₆(x) in terms of the known P₄(x) and P₅(x). See Problem 51 in Exercises 5.3.

Another formula, although not a recurrence relation, can generate the Legendre polynomials by differentiation:

(37)

The formula given in (37) first appeared in 1815 in the doctoral dissertation of the French mathematician Benjamin Olinde Rodrigues (1795–1851). So naturally, (37) is called Rodrigues’ formula. There are several such derivative formulas for generating polynomial solutions of differential equations with variable coefficients. See Problems 29 and 30 in Chapter 5 in Review.

5.3 Exercises Answers to selected odd-numbered problems begin on page ANS-13.

5.3.1 Bessel Functions

In Problems 1–6, use (1) to find the general solution of the given differential equation on (0, ∞).

1. x² y″ + xy′ + (x² − )y = 0
2. x² y″ + xy′ + (x² − 1)y = 0
3. 4x² y″ + 4xy′ + (4x² − 25)y = 0
4. 16x² y″ + 16xy′ + (16x² − 1)y = 0
5. xy″ + y′ + xy = 0
6. 

In Problems 7 and 8, use (12) to find the general solution of the given differential equation on the interval (0, ∞).

In Problems 9 and 10, use (18) to find the general solution of the given differential equation on (0, ∞).

In Problems 11 and 12, use the indicated change of variable to find the general solution of the given differential equation on the interval (0, ∞).

In Problems 13–20, use (20) to find the general solution of the given differential equation on the interval (0, ∞).

13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. Use the series in (7) to verify that I_ν (x) = i^−νJ_ν (ix) is a real function.
22. Assume that b in equation (20) can be pure imaginary; that is, b = βi, β > 0, i² = −1. Use this assumption to express the general solution of the given differential equation in terms of the modified Bessel functions I_n and K_n.

    (a) y″ − x² y = 0

    (b) xy″ + y′ − 7x³y = 0

In Problems 23–26, first use (20) to express the general solution of the given differential equation in terms of Bessel functions. Then use (26) and (27) to express the general solution in terms of elementary functions.

23. y″ + y = 0
24. x² y″ + 4xy′ + (x² + 2)y = 0
25. 16x² y″ + 32xy′ + (x⁴ − 12)y = 0
26. 4x² y″ − 4xy′ + (16x² + 3)y = 0

27. (a) Proceed as in Example 6 to show that

    

    [Hint: Write 2n + ν = 2(n + ν) − ν.]

    (b) Use the result in part (a) to derive (23).

28. Use the formula obtained in Example 6 along with part (a) of Problem 27 to derive the recurrence relation

    2ν J_ν (x) = x J_{ν + 1}(x) + xJ_ν − ₁(x).

In Problems 29 and 30, use (22) or (23) to obtain the given result.

29. 
30. J′₀(x) = J−₁(x) = −J₁(x)
31. 1. Proceed as on pages 296–297 to derive the elementary form of J_−1/2(x) given in (27).
    2. Use along with (26) and (27) in the recurrence relation in Problem 28 to express J_−3/2(x) in terms of sin x, cos x, and powers of x.
    3. Use a graphing utility to plot the graph of J_−3/2(x).
32. 1. Use the recurrence relation in Problem 28 to express J_3/2(x), J_5/2(x), and J_7/2(x) in terms of sin x, cos x, and powers of x.
    2. Use a graphing utility to plot the graphs of J_3/2(x), J_5/2(x), and J_7/2(x) in the same coordinate plane.
33. Use the solution of the aging spring equation , obtained in Example 5, to discuss the behavior of as in the three cases

    (a)

    (b)

    (c)

34. Show that y = x^1/2w(αx^3/2) is a solution of the given form of Airy’s differential equation whenever w is a solution of the indicated Bessel’s equation. [Hint: After differentiating, substituting, and simplifying, then let t = αx^3/2.]

    (a) y″ + α²xy = 0, x > 0; t²w″ + tw′ + (t² − )w = 0, t > 0

    (b) y″ − α²xy = 0, x > 0; t²w″ + tw′ − (t² + )w = 0, t > 0

35. Use the result in parts (a) and (b) of Problem 34 to express the general solution on (0, ∞) of each of the two forms of Airy’s equation in terms of Bessel functions.
36. Use Table 5.3.1 to find the first three positive eigenvalues and corresponding eigenfunctions of the boundary-value problem

    xy″ + y′ + λxy = 0,

    y(x), y′(x) bounded as x → 0¹, y(2) = 0.

    [Hint: By identifying λ = α², the DE is the parametric Bessel equation of order zero.]

37. 1. Use (20) to show that the general solution of the differential equation xy″ + λy = 0 on the interval (0, ∞) is

       

    2. Verify by direct substitution that y = is a particular solution of the DE in the case λ = 1.
38. Use I_ν(x) = i^−νJ_ν(ix) along with (7) and (8) to show that:

    (a) I_1/2 (x) =

    (b) I_−1/2(x) =

    (c) Use (16) to express K_1/2(x) in terms of elementary functions.

39. 1. Use the first formula in (31) to find the spherical Bessel functions j₁(x), j₂(x), and j₃(x).
    2. Use a graphing utility to plot the graphs of j₁(x), j₂(x), and j₃(x) in the same coordinate plane.
40. 1. Use the second formula in (31) to find the spherical Bessel functions y₁(x), y₂(x), and y₃(x).
    2. Use a graphing utility to plot the graphs of y₁(x), y₂(x), and y₃(x) in the same coordinate plane.
41. If n is an integer, use the substitution to show that the differential equation

    (38)

    becomes

    (39)

42. 1. In Problem 41, find the general solution of the differential equation in (39) on the interval (0, ∞).
    2. Use part (a) to find the general solution of the differential equation in (38) on the interval (0, ∞).
    3. Use part (b) to express the general solution of (38) in terms of the spherical Bessel functions of the first and second kind defined in (31).

Mathematical Model

43. Cooling Fin A cooling fin is an outward projection from a mechanical or electronic device from which heat can be radiated away from the device into the surrounding medium (such as air). See the accompanying photo. An annular, or ring-shaped, cooling fin is normally used on cylindrical surfaces such as a circular heating pipe. See FIGURE 5.3.7. In the latter case, let r denote the radial distance measured from the center line of the pipe and T(r) the temperature within the fin defined for r₀ ≤ r ≤ r₁. It can be shown that T(r) satisfies the differential equation

    ,

    where a² is a constant and T_m is the constant air temperature. Suppose r₀ = 1, r₁ = 3, and T_m = 70. Use the substitution w(r) = T(r) − 70 to show that the solution of the given differential equation subject to the boundary conditions

    

    is ,

    where I₀(x) and K₀(x) are the modified Bessel functions of the first and second kind. You will also have to use the derivatives given in (25) on page 296.

    

    

44. Solve the differential equation in Problem 43 when the boundary conditions are

    .

Computer Lab Assignments

45. (a) Use the general solution given in Example 5 to solve the IVP

    

    Also use J′₀(x) = −J₁(x) and Y′₀(x) = −Y₁(x) along with Table 5.3.1 or a CAS to evaluate coefficients.

    (b) Use a CAS to graph the solution obtained in part (a) for 0 ≤ t < ∞.

46. (a) Use the general solution obtained in Problem 35 to solve the IVP

    

    Use a CAS to evaluate coefficients.

    (b) Use a CAS to graph the solution obtained in part (a) for 0 ≤ t ≤ 200.

47. Column Bending Under Its Own Weight A uniform thin column of length L, positioned vertically with one end embedded in the ground, will deflect, or bend away, from the vertical under the influence of its own weight when its length or height exceeds a certain critical value. It can be shown that the angular deflection θ(x) of the column from the vertical at a point P(x) is a solution of the boundary-value problem

    

    where E is Young’s modulus, I is the cross-sectional moment of inertia, δ is the constant linear density, and x is the distance along the column measured from its base. See FIGURE 5.3.8.

    

    The column will bend only for those values of L for which the boundary-value problem has a nontrivial solution.

    1. Restate the boundary-value problem by making the change of variables t = L − x. Then use the results of a problem earlier in this exercise set to express the general solution of the differential equation in terms of Bessel functions.
    2. Use the general solution found in part (a) to find a solution of the BVP and an equation that defines the critical length L, that is, the smallest value of L for which the column will start to bend.
    3. With the aid of a CAS, find the critical length L of a solid steel rod of radius r = 0.05 in., δg = 0.28 A lb/in., E = 2.6 × 10⁷ lb/in.² , A = πr², and I = πr⁴.
48. Buckling of a Thin Vertical Column In Example 4 of Section 3.9 we saw that when a constant vertical compressive force, or load, P was applied to a thin column of uniform cross section and hinged at both ends, the deflection y(x) is a solution of the boundary-value problem:

    

    1. If the bending stiffness factor EI is proportional to x, then EI(x) = kx, where k is a constant of proportionality. If EI(L) = kL = M is the maximum stiffness factor, then k = M/L and so EI(x) = Mx/L. Use the information in Problem 37 to find a solution of

       

       if it is known that is not zero at x = 0.

    2. Use Table 5.3.1 to find the Euler load P₁ for the column.
    3. Use a CAS to graph the first buckling mode y₁(x) corresponding to the Euler load P₁. For simplicity assume that c₁ = 1 and L = 1.
49. Pendulum of Varying Length For the simple pendulum described on page 194 of Section 3.11, suppose that the rod holding the mass m at one end is replaced by a flexible wire and that the wire is fed by a pulley through the horizontal support at point O in FIGURE 5.3.9. In this manner, while it is in motion in a vertical plane, the mass m can be raised or lowered. In other words, the length l(t) of the pendulum varies with time. Under the same assumptions leading to equation (6) in Section 3.11, it follows from (1) in Chapter 3 in Review that the differential equation for the displacement angle θ(t) is now

    

    1. If l increases at a constant rate v and l(0) = l₀, then show that a linearization of the foregoing differential equation is

       (40)

    2. Make the change of variables x = (l₀ + vt)/v and show that (40) becomes

       

    3. Use part (b) and (20) to express the general solution of equation (40) in terms of Bessel functions.
    4. Use the general solution obtained in part (c) to solve the initial-value problem consisting of equation (40) and the initial conditions θ(0) = θ₀, θ′(0) = 0. [Hints: To simplify calculations use a further change of variable u = Also, recall (22) holdsfor both J₁(u) and y₁(u). Finally, the identity

       J₁(u)y₂(u) − J₂(u)y₁(u) = −

       will be helpful.]

    5. Use a CAS to graph the solution θ(t) of the IVP in part (d) when l₀ = 1 ft, θ₀ = radian, and v = ft/s. Experiment with the graph using different time intervals such as [0, 10], [0, 30], and so on.
    6. What do the graphs indicate about the displacement angle θ(t) as the length l of the wire increases with time?

5.3.2 Legendre Functions

50. 1. Use the explicit solutions y₁(x) and y₂(x) of Legendre’s equation given in (33) and the appropriate choice of c₀ and c₁ to find the Legendre polynomials P₆(x) and P₇(x).
    2. Write the differential equations for which P₆(x) and P₇(x) are particular solutions.
51. Use the recurrence relation (36) and P₀(x) = 1, P₁(x) = x, to generate the next six Legendre polynomials.
52. Show that the differential equation

    

    can be transformed into Legendre’s equation by means of the substitution x = cos θ.

53. Find the first three positive values of λ for which the problem

    (1 − x²)y″ − 2xy′ + λy = 0,

    y(0) = 0, y(x), y′(x) bounded on [−1, 1]

    has nontrivial solutions.

54. The differential equation

    

    is known as the associated Legendre equation. When m = 0 this equation reduces to Legendre’s equation (2). A solution of the associated equation is

    

    where P_n(x), n = 0, 1, 2, . . . are the Legendre polynomials given in (34). The solutions for . . ., are called associated Legendre functions.

    1. Find the associated Legendre functions
    2. What can you say about when m is an even nonnegative integer?
    3. What can you say about when m is a nonnegative integer and
    4. Verify that satisfies the associated Legendre equation when and

Computer Lab Assignments

55. For purposes of this problem, ignore the list of Legendre polynomials given on page 299 and the graphs given in Figure 5.3.6. Use Rodrigues’ formula (37) to generate the Legendre polynomials P₁(x), P₂(x), . . . , P₇(x). Use a CAS to carry out the differentiations and simplifications.
56. Use a CAS to graph P₁(x), P₂(x), . . . , P₇(x) on the closed interval [−1, 1].
57. Use a root-finding application to find the zeros of P₁(x), P₂(x), . . . , P₇(x). If the Legendre polynomials are built-in functions of your CAS, find the zeros of Legendre polynomials of higher degree. Form a conjecture about the location of the zeros of any Legendre polynomial P_n(x), and then investigate to see whether it is true.

*When we replace x by |x|, the series given in (7) and (8) converge for 0 < |x| < ∞.