WRITING PROJECT. Power Series. Write a review (2-3 pages)

Apply the power series method. Do this by hand, not by a CAS, so that you get a feel for the method, e.g., why a series may terminate, or has even powers only, or has no constant or linear terms, etc. Show the details of your work.

Apply the power series method. Do this by hand, not by a CAS, so that you get a feel for the method, e.g., why a series may terminate, or has even powers only, or has no constant or linear terms, etc. Show the details of your work.

Apply the power series method. Do this by hand, not by a CAS, so that you get a feel for the method, e.g., why a series may terminate, or has even powers only, or has no constant or linear terms, etc. Show the details of your work.

Apply the power series method. Do this by hand, not by a CAS, so that you get a feel for the method, e.g., why a series may terminate, or has even powers only, or has no constant or linear terms, etc. Show the details of your work.

Apply the power series method. Do this by hand, not by a CAS, so that you get a feel for the method, e.g., why a series may terminate, or has even powers only, or has no constant or linear terms, etc. Show the details of your work.

Apply the power series method. Do this by hand, not by a CAS, so that you get a feel for the method, e.g., why a series may terminate, or has even powers only, or has no constant or linear terms, etc. Show the details of your work.

Apply the power series method. Do this by hand, not by a CAS, so that you get a feel for the method, e.g., why a series may terminate, or has even powers only, or has no constant or linear terms, etc. Show the details of your work.

Apply the power series method. Do this by hand, not by a CAS, so that you get a feel for the method, e.g., why a series may terminate, or has even powers only, or has no constant or linear terms, etc. Show the details of your work.

Apply the power series method. Do this by hand, not by a CAS, so that you get a feel for the method, e.g., why a series may terminate, or has even powers only, or has no constant or linear terms, etc. Show the details of your work.

Apply the power series method. Do this by hand, not by a CAS, so that you get a feel for the method, e.g., why a series may terminate, or has even powers only, or has no constant or linear terms, etc. Show the details of your work.

Solve the initial value problems by a power series. Graph the partial sum s of the powers up to and including x 5 . Find the value of s (5 digits) at Xl'

Solve the initial value problems by a power series. Graph the partial sum s of the powers up to and including x 5 . Find the value of s (5 digits) at Xl'

Solve the initial value problems by a power series. Graph the partial sum s of the powers up to and including x 5 . Find the value of s (5 digits) at Xl'

Solve the initial value problems by a power series. Graph the partial sum s of the powers up to and including x 5 . Find the value of s (5 digits) at Xl'

Solve the initial value problems by a power series. Graph the partial sum s of the powers up to and including x 5 . Find the value of s (5 digits) at Xl'

Solve the initial value problems by a power series. Graph the partial sum s of the powers up to and including x 5 . Find the value of s (5 digits) at Xl'

WRITING PROJECT. Power Series. Write a review (2-3 pages) on power series as they are discussed in calculus, using your own formulation and examplesdo not just copy passages from calculus texts.

LITERATURE PROJECT. Maclaurin Series. Collect Maclaurin series of the functions known from calculus and arrange them systematically in a list that you can use for your work.

Determine the radius of convergence. (Show the details.)

Determine the radius of convergence. (Show the details.)

Determine the radius of convergence. (Show the details.)

Determine the radius of convergence. (Show the details.)

Determine the radius of convergence. (Show the details.)

Determine the radius of convergence. (Show the details.)

Determine the radius of convergence. (Show the details.)

Determine the radius of convergence. (Show the details.)

Determine the radius of convergence. (Show the details.)

Determine the radius of convergence. (Show the details.)

Determine the radius of convergence. (Show the details.)

Determine the radius of convergence. (Show the details.)

Thi~ is often convenient or nece~sary in the power series method. Shift the index so that the power under the summation sign is xS. Check by writing the first few terms explicitly. Also determine the radius of convergence R.

Thi~ is often convenient or nece~sary in the power series method. Shift the index so that the power under the summation sign is xS. Check by writing the first few terms explicitly. Also determine the radius of convergence R.

Thi~ is often convenient or nece~sary in the power series method. Shift the index so that the power under the summation sign is xS. Check by writing the first few terms explicitly. Also determine the radius of convergence R.L p x p + 4 p~l (p + l)!

Find a power series solution in powers of x. (Show the details of your work.)

Find a power series solution in powers of x. (Show the details of your work.)

Find a power series solution in powers of x. (Show the details of your work.)

Find a power series solution in powers of x. (Show the details of your work.)

Find a power series solution in powers of x. (Show the details of your work.)

Find a power series solution in powers of x. (Show the details of your work.)

Find a power series solution in powers of x. (Show the details of your work.)

Find a power series solution in powers of x. (Show the details of your work.)

TEAM PROJECT. Properties from Power Series. In the next sections we shall define new functions (Legendre functions. etc.) by power series. deriving properties of the functions directly from the series. To understand this idea. do the same for functions familiar from calculus. using Maclaurin series. (a) Show that cosh x + sinh x = eX. Show that cosh x > 0 for all x. Show that eX ;:;:; e-x for all x;:;:; O. (b) Derive the differentiation formulas for eX. cos x, sinx. 11(1 - x) and other functions of your choice. Show that (cos xl" = -cos x. (cosh :d' = cosh x. Consider integration similarly. (c) What can you conclude if a series contains only odd powers? Only even powers? No constant tenn? If all its coefficients are positive? Give examples. (d) What properties of cos x and sin x are lIot obvious from the Maclaurin series? What properties of other functions?

CAS EXPERIMENT. Information from Graphs of Partial Sums. In connection with power series in numerics we use partial sums. To get a feel for the accuracy for various x. experiment with sin x and graphs of partial sums of the Maclaurin series of an increasing number of terIllS, describing qualitatively the "breakaway points" of these graphs from the graph of sin x. Consider other examples of your own choice.

Verify that the polynomials in (11') satisfy Legendre's equation.

Derive (11 ') from (11).

Obtain P6 and P7 from (11).

Convergence) Show that for any 11 for which (6) or (7) does nol reduce to a polynomial, the series has radius of convergence 1.

Legendre function Qo(x) for n = 0) Show that (6)with 11 = 0 gives Yl(X) = Po(x) = I and (7) gives Y2(X) = x + - x3 + x 5 + ... 3! 5! x 3 X5 I I + x =x+ +-+=-In--. 3 5 2 I-x Verify this by solving (\) with /I = 0, setting z = y' and separating variables.

(Legendre function -Ql(X) for II = 1) Show that (7) with 11 = I gives )"2(X) = PI(x) = x and (6) gives )"I(X) = -Ql(X) (the minus sign in the notation being conventional), )"I(X) = I - 3 5 I - x (r + .~ + .~5 + ... ) I I + x = 1- -xln-- 2 I - x

(ODE) Find a solution of (a 2 - x 2 )y" - 2xy' + n(/1 + l)y = 0. a *' 0. by reduction to the Legendre equation

[Rodrigues's formula (12)]2 Applying the binomial theorem to (X2 - I)n, differentiating it 11 times term by term. and comparing the result with (II), show that (12)

(Rodrigues's formula) Obtain (II ') from (12).

Graph P2 (x) ... PIO(.\) on common axes. For what \" (approximately) and II = 2 ... " 10 is Ipn(x)1

From what /I on will your CAS no longer produce faithful graphs of P n(x)? Why?

Graph Qo(x), QI (x), and some further Legendre functions.

Substitute asxs + a s+ IXs + 1 + as+2xs+2 into Legendre's equation and obtain the coefficient recursion (4).

TEAM PROJECT. Generating Functions. Generating functions playa significant role in modem applied mathematics (see [GR5]). The idea is simple. If we want to study a certain sequence (fn(x and can find a function "" C(u, x) = L fn(x)u n, n=O we may obtain properties of (f ,,(x from those of C. which ""generates" this sequence and is called a generating function of the sequence. (a) Legendre polynomials. Show that (13) is a generating function of the Legendre polynomials. Hint: Start from the binomial expansion of 11"\ 1 - v. then set v = 2xlI - u2 multiply the powers of 2m - u2 out. collect all the terms involving un, and verify that the slim of these terms is Pn(x)u n. (b) Potential theory. Let Al and A2 be two points in space (Fig. 105, r2 > 0). Using (13), show that 1 r Vr12 + r22 - 2rlr2 cos e This formula has applications in potential theory. (Qlr is the electrostatic potential at A2 due to a charge Q located at Al . And the series expresses I1r in terms of the distances of Al and A2 from any origin o and the angle e between the segments OAI and OA2 ) Fig. 105. T earn Project 14 (c) Further applications of (13). Show that Pn(l) = I, Pn( -I) = (-It'. P2n+ 1(0) = 0, and P2n(O) = (-I)n'I'3'" (211 - 1)/[2,4", (2n)]. (d) Bonnet's recursion.3 Differentiating (\3) with respect to u, using (13) in the resulting formula, and comparing coefficients of un, obtain the Bonnet recursion (14) (II + l)Pn+l(x) = (211 + I)xP,/:r) - IlPn_I(X), where Il = I, 2, .... This formula is useful for computations, the loss of significant digits being small (except near zeros). Try ( 14) out for a few computations of your own choice.

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

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) (x + 2)2)''' - 2)' = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.)xv" + 51" + xy = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.)2xy" + (3 - 4.1.'lY' + (2x - 3)y 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.)x 2 )"" + -1-.1:\" + (x 2 + 2)" = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.)4.1.')," + 2/ + y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.)(x + 3)2)''' - 9(x + 3),' + 25y o

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.)xy" - }' = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.)x/' + (2x + 1)/ + (x + 1), = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.)1'2)''' + 2x3 ),' + (x 2 - 2)y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) (x 2 + f)Y" + (4x + 2),' + 2)" = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.)x 2y" + 6xy' + (4x 2 + 6)y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.)2x)"" - (8x - 1)y' + (8x - 2).\' 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) .1.'y" + y' - xy = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) (x - 4)2)''' - (x - 4)y' - 35y o

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.)x 2y" + 4xy' - (x2 - 2)y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.)v" + (x - 6)' = 0

TEAM PROJECT. Hypergeometric Equation, Series, and Function. Gauss' 5 hypergeometric ODE5

Find a general solution in terms of hypergeometric functions.x(\ - x)''' + (! - 2x)), , - !y = 0

Find a general solution in terms of hypergeometric functions.2x(l - x," - (1 + 6x)y' - 2y = 0

Find a general solution in terms of hypergeometric functions.x(1 - x),," + h' + 2.\' = 0

Find a general solution in terms of hypergeometric functions.3[( I + t)y + t)' - )' = 0

Find a general solution in terms of hypergeometric functions.2(2 - 5t + 6)5; + (2t - 3).} 8y = 0

Find a general solution in terms of hypergeometric functions.4(t2 - 3t + 2)5: - 2." + Y o

(Convergence) Show that the series in (I I) converges for all x. Why is the convergence very rapid?

(Approximation) Show that for small Ixl we have 10 = I - 0.25x2. From this compute 10(.'r) for x = O. 0.1. 0.2 ..... 1.0 and determine the error by using Table Al in App. 5 or your CAS.

("'Large" values) Using 04), compute 10lx) for x = 1.0, 2.0. 3.0 ..... 8.0, determine the error by Table Al or your CAS. and comment.

(Zeros) Compute the fIrst four positive zeros of 10(.r) and 11(x) from (14). Determine the error and comment.

Using the indicated substitution~. find a general solution in temlS of 1 v and 1 -v or indicate when this is not possible. (This is just a sample of various ODEs reducible to Bessel's equation. Some more follow in the next problem set. Show the details of your work.) (ODE "ith two parameters) x 2)"" + xy' + (A2x 2 - JJ2)y = 0 lAx = ;::)

Using the indicated substitution~. find a general solution in temlS of 1 v and 1 -v or indicate when this is not possible. (This is just a sample of various ODEs reducible to Bessel's equation. Some more follow in the next problem set. Show the details of your work.)x 2 /' + xy' + (x 2 - -W)Y = 0

Using the indicated substitution~. find a general solution in temlS of 1 v and 1 -v or indicate when this is not possible. (This is just a sample of various ODEs reducible to Bessel's equation. Some more follow in the next problem set. Show the details of your work.)x 2)''' + xy' + !(x - JJ2)y = 0 CVx = z)

Using the indicated substitution~. find a general solution in temlS of 1 v and 1 -v or indicate when this is not possible. (This is just a sample of various ODEs reducible to Bessel's equation. Some more follow in the next problem set. Show the details of your work.) (2x + 1 )2y" + 2(2x + 1)/ + 16x( t + l)y = 0 (2x + I = ::.;)

Using the indicated substitution~. find a general solution in temlS of 1 v and 1 -v or indicate when this is not possible. (This is just a sample of various ODEs reducible to Bessel's equation. Some more follow in the next problem set. Show the details of your work.)x/' - / + 4ry = 0 (\. = XII, 2t = z)

Using the indicated substitution~. find a general solution in temlS of 1 v and 1 -v or indicate when this is not possible. (This is just a sample of various ODEs reducible to Bessel's equation. Some more follow in the next problem set. Show the details of your work.)x 2 )"" + x.v' + !(x2 - I)), = 0 lx = 2::.;)

Using the indicated substitution~. find a general solution in temlS of 1 v and 1 -v or indicate when this is not possible. (This is just a sample of various ODEs reducible to Bessel's equation. Some more follow in the next problem set. Show the details of your work.)xy" + (2v + l)y' + xy = 0 (y = x-Vu)

Using the indicated substitution~. find a general solution in temlS of 1 v and 1 -v or indicate when this is not possible. (This is just a sample of various ODEs reducible to Bessel's equation. Some more follow in the next problem set. Show the details of your work.)x2)"" + xy' + 4(x4 - JJ2)y = 0 (x2 = z)

Using the indicated substitution~. find a general solution in temlS of 1 v and 1 -v or indicate when this is not possible. (This is just a sample of various ODEs reducible to Bessel's equation. Some more follow in the next problem set. Show the details of your work.)... 2y" + xy' + 9(x6 - v 2 ' = 0 (x 3 = z)

Using the indicated substitution~. find a general solution in temlS of 1 v and 1 -v or indicate when this is not possible. (This is just a sample of various ODEs reducible to Bessel's equation. Some more follow in the next problem set. Show the details of your work.)y" + (e2x - ~)y = 0 (eX = .::.)

Using the indicated substitution~. find a general solution in temlS of 1 v and 1 -v or indicate when this is not possible. (This is just a sample of various ODEs reducible to Bessel's equation. Some more follow in the next problem set. Show the details of your work.)xy" + y = 0 l)' = Vx u. 2Vx = z)

Using the indicated substitution~. find a general solution in temlS of 1 v and 1 -v or indicate when this is not possible. (This is just a sample of various ODEs reducible to Bessel's equation. Some more follow in the next problem set. Show the details of your work.)16x2 )''' + 8xy' + (x1/ 2 + :a)y = 0 (y = xl/4 u. X1l4 = ::.;)

Using the indicated substitution~. find a general solution in temlS of 1 v and 1 -v or indicate when this is not possible. (This is just a sample of various ODEs reducible to Bessel's equation. Some more follow in the next problem set. Show the details of your work.)36x2,," + 18x\" + Vx V = 0 (y =. x 1l4u, ixi/4 = z)

Using the indicated substitution~. find a general solution in temlS of 1 v and 1 -v or indicate when this is not possible. (This is just a sample of various ODEs reducible to Bessel's equation. Some more follow in the next problem set. Show the details of your work.)x 2 )''' + xy' + Vx y = 0 (4t1l4 = z)

Using the indicated substitution~. find a general solution in temlS of 1 v and 1 -v or indicate when this is not possible. (This is just a sample of various ODEs reducible to Bessel's equation. Some more follow in the next problem set. Show the details of your work.)x 2)"" + !xy' + Vx y = 0 ly = x 2/5u. 4X1l4 = .::.)

Using the indicated substitution~. find a general solution in temlS of 1 v and 1 -v or indicate when this is not possible. (This is just a sample of various ODEs reducible to Bessel's equation. Some more follow in the next problem set. Show the details of your work.) x 2/' + (I - 2v)xy' + v 2(x2v + 1 - v2)y = 0 (y = t"u, XV = ::.;)

Use the powerful formulas (24) to do Probs. 21-28. (Show the details of your work.)(Derivatives) Show that l~(x) = - 11(x), l~(x) = 10(x) - 11(X)/X, l~(x) = Ml1(x) - 13(x)l.

Use the powerful formulas (24) to do Probs. 21-28. (Show the details of your work.)(Interlacing ofzeros) Using (24) and Rolle's theorem, show that between two consecutive zeros of 10(x) there is precisely one zero of 11 (x).

Use the powerful formulas (24) to do Probs. 21-28. (Show the details of your work.)(Interlacing ofzeros) Using (24) and Rolle's theorem. show that between any two consecutive positive zeros of In(x) there i~ precisely one J:ero of l,,+I(X).

Use the powerful formulas (24) to do Probs. 21-28. (Show the details of your work.)(Bessel's equation) Derive (I) from (24).

Use the powerful formulas (24) to do Probs. 21-28. (Show the details of your work.) (Basic integral formulas) Show that

Use the powerful formulas (24) to do Probs. 21-28. (Show the details of your work.) (Integration) Evaluate I x- 1 14 (x) dx. (Use Prob. 25; integrate by parts.)

Use the powerful formulas (24) to do Probs. 21-28. (Show the details of your work.) (Integration) Show that Ix 2 10 (x) dx = x211 (x) + x10(x) - Ilo(X) dx. (The last integral is nonelementary; tables exist, e.g. in Ref. [A13J in App. L)

Use the powerful formulas (24) to do Probs. 21-28. (Show the details of your work.) (Integration) Evaluate I15(X) dx.

(Elimination of first derivative) Show that y = ltV with vex) = exp (-~ J p(x) dx) gives from the ODE y" + p(x)y' + q(x)y = 0 the ODE no longer containing the first derivative. Show that for the Bessel equation the substitution is y = UX-1I2 and gives

(Elementary Bessel functions) Derive (25) in Theorem 4 from (27).

CAS EXPERIMENT. Change of Coefficient. Find and graph (on common axes) the solutions of .v" + h-1 y' + Y = O .1'(0) = I. y' (0) = O. for k = 0, 1, 2 ... '. 10 (or as far as you get useful graphs). For what k do you get elementary functions? Why? Try for noninteger k. particularly between 0 and 2. to see the continuous change of the curve. Describe the change of the location of the zeros and of the extrema as k increases from O. Can you interpret the ODE as a model in mechanics, thereby explaining your observations?

TEAM PROJECT. Modeling a Vibrating Cable (Fig. 108). A flexible cable, chain, or rope of length L and density (mass per unit length) p is fixed at the upper end (x = 0) and allowed to make small vibrations (small angles a in the horizontal displacement u(x. t), t = time) in a vertical plane. (a) Show the following. The weight of the cable below a point x is W(x) = pg(L - x). The restoring force is F(x) = W sin a = Wu.". Ux = (JuliJx. The difference in force between x and x + D.X is D.x (Wu~')x' Newton's second law now gives p .1x IItt = .lx pg[(L - X)lI"Jx. For the expected periodic motion u(x, t) = .r(x) cos (wt + 8) the model of the problem is the ODE (L - x).r" - y' + A\ = O. (b) Transform this ODE to~; + ,1'-1" + Y = O. . " = dylds. s = 2A::,1I2. ::. = L - x. so that the solution is y(x) = 10 (2wV(L - >;)Ig).(c) Conclude that possible frequencies wl27f are those for which s = 2wv Llg is a zero of 10 , The con-esponding solutions are called normal modes. Figure 108 shows the first of them. What does the second normal mode look like? The third? What is the frequency (cycles/min) of a cable of length 2 m? Of length 10 m? (c) Conclude that possible frequencies wl27f are those for which s = 2wv Llg is a zero of 10 , The con-esponding solutions are called normal modes. Figure 108 shows the first of them. What does the second normal mode look like? The third? What is the frequency (cycles/min) of a cable of length 2 m? Of length 10 m?

CAS EXPERIMENT. Bessel Functions for Large x. (a) Graph l,,(x) for 11 = 0, ... , 5 on common axes. (b) Experiment with (14) for integer II. Using graphs. find out from which x = Xn on the curves of (II) and (14) practically coincide. How does Xn change with n? (c) What happens in (b) if II = ::!::~? (Our usual notation in this case would be v.) (d) How does the en-or of (14) behave as function of x for fixed II? [En-or = exact value minus approximation (14).1 (e) Show from the graphs that 10 (x) has extrema where 11(x) = O. Which formula proves this? Find further relations between zeros and extrema . (f) Raise and answer questions of your own. for instance. on the zeros of 10 and 11 , How accurately are they obtained from (14)?

Using the indicated substinl!ions, find a general solution in terms of J v and Y . Indicate whether you could also use J- v instead of Yv ' (Show the details of your work.)

Using the indicated substinl!ions, find a general solution in terms of J v and Y . Indicate whether you could also use J- v instead of Yv ' (Show the details of your work.)

Using the indicated substinl!ions, find a general solution in terms of J v and Y . Indicate whether you could also use J- v instead of Yv ' (Show the details of your work.)

Using the indicated substinl!ions, find a general solution in terms of J v and Y . Indicate whether you could also use J- v instead of Yv ' (Show the details of your work.)

Using the indicated substinl!ions, find a general solution in terms of J v and Y . Indicate whether you could also use J- v instead of Yv ' (Show the details of your work.)

Using the indicated substinl!ions, find a general solution in terms of J v and Y . Indicate whether you could also use J- v instead of Yv ' (Show the details of your work.)

Using the indicated substinl!ions, find a general solution in terms of J v and Y . Indicate whether you could also use J- v instead of Yv ' (Show the details of your work.)

Using the indicated substinl!ions, find a general solution in terms of J v and Y . Indicate whether you could also use J- v instead of Yv ' (Show the details of your work.)

Using the indicated substinl!ions, find a general solution in terms of J v and Y . Indicate whether you could also use J- v instead of Yv ' (Show the details of your work.)

Using the indicated substinl!ions, find a general solution in terms of J v and Y . Indicate whether you could also use J- v instead of Yv ' (Show the details of your work.)

(Hankel functions) Show that the Hankel functions ( 10) form a basis of solutions of Bessel's equation for any v.

CAS EXPERIMENT. Bessel Functions for Large x. It can be shown that for large x. (11) Yn(x) - v'2/( 7TX) sin (x - ! 1l7T - ~7T) with - defined as in (14) of Sec. 5.5. (a) Graph Yn(x) for 11 = O .... , 5 on common axes. Are there relations between zeros of one function and extrema of another? For what functions? (b) Find out from graphs from which x = Xn on the curves of (8) and (11) (both obtained from your CAS) practically coincide. How does Xn change with 11? (c) Calculate the first ten Leros Xm ' In = I, ... , 10, of Yo(x) from your CAS and from (II). How does the error behave as 171 increases? (d) Do (c) for Yl(X) and Y2(x). How do the errors compare to those in (c)?

Modified Bessel functions of the first kind of order P are defined by Iv(x) = i-vJ,,(ix), i = \1"="1. Show that Iv satisfies the ODE (12) x 2y" + xy' - (x2 + v 2 )y = 0 and has the representation cc x2m.+v (13) [,Jx) = L 22m+vm! rem + v + I)

(Modified Bessel functions I.,) Show that I,,(x) is real for all real x (and real v), 1 v (x) "* 0 for all real x "* 0, and Ln(x) = In(x). where n is any integer.

Modified Bessel functions of the third kind (sometimes called of the second kind) are defined by the formula (14) below. Show that they satisfy the ODE (12). 7r (14) KJI:) = . [Lv(.\) - IJr)]

(Proof of Theorem 1) CalTY out the details in Cases 3 and 4.

Normalization of eigenfunctions Ym of (I), (2) means that we multiply Ym by a nonzero constant em such that emYm has norm I. Show that 2m = ey", with any e * 0 is an eigenfunction for the eigenvalue corresponding to vm -

(Change of x) Show that if the functions Yo(x), YI(X), ... form an orthogonal set on an interval a ~ x ~ b (with rex) = I), then the functions )'o(er + k), )'l(er + k), ... , e > 0, form an orthogonal set on the interval (a - k)/e ~ I ~ (b - k)!c.

(Change of x) Using Prob. 3, derive the orthogonality of I, cos wx, sin wx, cos 2wx. sin 2wx. ... on -1 ~ x ~ I (r(x) = 1) from that of 1, cos x, sin x, cos 2x, sin 2x, ... on -w ~ x ~ rr.

(Legendre polynomials) Show that the functions P,,{cos 6), n = 0, I, ... , form an orthogonal set on the interval 0 ~ e ~ rr with respect to the weight function sin e.

(Tranformation to Sturm-Liom iIIe form) Show that Y" + fy' + (g + Ah)' = 0 takes the form (I) if you set p = exp (If dx), q = pg, r = hp. Why would you do such a transformation?

Write the given ODE in the form (I) if it is in a different form. (Use Prob. 6.) Find the eigenvalues and eigenfunctions. Verify orthogonality. (Show the details of your work.)

Write the given ODE in the form (I) if it is in a different form. (Use Prob. 6.) Find the eigenvalues and eigenfunctions. Verify orthogonality. (Show the details of your work.)

Write the given ODE in the form (I) if it is in a different form. (Use Prob. 6.) Find the eigenvalues and eigenfunctions. Verify orthogonality. (Show the details of your work.)

Write the given ODE in the form (I) if it is in a different form. (Use Prob. 6.) Find the eigenvalues and eigenfunctions. Verify orthogonality. (Show the details of your work.)

Write the given ODE in the form (I) if it is in a different form. (Use Prob. 6.) Find the eigenvalues and eigenfunctions. Verify orthogonality. (Show the details of your work.)

Write the given ODE in the form (I) if it is in a different form. (Use Prob. 6.) Find the eigenvalues and eigenfunctions. Verify orthogonality. (Show the details of your work.)

Write the given ODE in the form (I) if it is in a different form. (Use Prob. 6.) Find the eigenvalues and eigenfunctions. Verify orthogonality. (Show the details of your work.)

Write the given ODE in the form (I) if it is in a different form. (Use Prob. 6.) Find the eigenvalues and eigenfunctions. Verify orthogonality. (Show the details of your work.)

Write the given ODE in the form (I) if it is in a different form. (Use Prob. 6.) Find the eigenvalues and eigenfunctions. Verify orthogonality. (Show the details of your work.)

Write the given ODE in the form (I) if it is in a different form. (Use Prob. 6.) Find the eigenvalues and eigenfunctions. Verify orthogonality. (Show the details of your work.)

Write the given ODE in the form (I) if it is in a different form. (Use Prob. 6.) Find the eigenvalues and eigenfunctions. Verify orthogonality. (Show the details of your work.)

Write the given ODE in the form (I) if it is in a different form. (Use Prob. 6.) Find the eigenvalues and eigenfunctions. Verify orthogonality. (Show the details of your work.)

Write the given ODE in the form (I) if it is in a different form. (Use Prob. 6.) Find the eigenvalues and eigenfunctions. Verify orthogonality. (Show the details of your work.)

TEAM PROJECT. Special Functions. Orthogonal polynomials playa great role in applications. For this reason, Legendre polynomials and various other orthogonal polynomials have been studied extensively; see Refs. [GR1], [GRIO] in App. 1. Consider some of the most important ones as follows. (a) Chebyshev pol)nomiaIslO of the first and second kind are defined by Tn(x) = cos (n arccos x) sin [(n + I) arccos x] Un(x) = ~ respectively, where 11 = O. 1, .. '. Shuw that To = 1, Uo = 1, Show that the Chebyshev polynomials Tn(x) are orthogonal on the interval - I ~ x ~ I with respect to the weight function rex) = 1I~. (Hint. To evaluate the integral, set arccos x = e.) Verify that T1l(x), 11 = 0, I, 2. 3, satisfy the Chebyshev equation (1 - x 2 )y" - xy' + n2 y = O. (b) Orthogonality on an infinite interval: Laguerre polynomialsll are defined by Lo = I, and n = 1,2,'" Show that Prove that the Laguerre polynomials are orthogonal on the positive axis 0 ~ x < :x; with respect to the weight function /"p:) = e-x . Hint. Since the highest power in Lm is x"', it suffices to show that f e-xxkLn dx = 0 for k < n. Do this by k integrations by parts.

Showing the details of your calculations, develop:7x4 - 6x2

Showing the details of your calculations, develop:(x + 1)2

Showing the details of your calculations, develop:

Showing the details of your calculations, develop:

Prove that if f(x) in Example 2 is even [that is, f(x) = f( - x)], its series contains only P m(x) with even 111.

Find and graph (on common axes) the partial sums up to that S"'o whose graph practically coincides with that of/ex) within graphical accuracy. State what 1Il0 is. On what does the size of IIlO seem to depend?f(x) sin TTX

Find and graph (on common axes) the partial sums up to that S"'o whose graph practically coincides with that of/ex) within graphical accuracy. State what 1Il0 is. On what does the size of IIlO seem to depend?f(x) sin 27TX

Find and graph (on common axes) the partial sums up to that S"'o whose graph practically coincides with that of/ex) within graphical accuracy. State what 1Il0 is. On what does the size of IIlO seem to depend?f(x) cos TTX

Find and graph (on common axes) the partial sums up to that S"'o whose graph practically coincides with that of/ex) within graphical accuracy. State what 1Il0 is. On what does the size of IIlO seem to depend?f(x) cos 27TX

Find and graph (on common axes) the partial sums up to that S"'o whose graph practically coincides with that of/ex) within graphical accuracy. State what 1Il0 is. On what does the size of IIlO seem to depend?f(x) cos 37TX

Find and graph (on common axes) the partial sums up to that S"'o whose graph practically coincides with that of/ex) within graphical accuracy. State what 1Il0 is. On what does the size of IIlO seem to depend?f(x) eX

Find and graph (on common axes) the partial sums up to that S"'o whose graph practically coincides with that of/ex) within graphical accuracy. State what 1Il0 is. On what does the size of IIlO seem to depend? f(x) e- x

Find and graph (on common axes) the partial sums up to that S"'o whose graph practically coincides with that of/ex) within graphical accuracy. State what 1Il0 is. On what does the size of IIlO seem to depend?f(x) = 1/(1 + x 2)

Find and graph (on common axes) the partial sums up to that S"'o whose graph practically coincides with that of/ex) within graphical accuracy. State what 1Il0 is. On what does the size of IIlO seem to depend?f(x) = 10(aO,1x). where aO,1 is the first positive zero of 10

Find and graph (on common axes) the partial sums up to that S"'o whose graph practically coincides with that of/ex) within graphical accuracy. State what 1Il0 is. On what does the size of IIlO seem to depend?f(x) = 10(aO,2x), where aO,2 is the second positive zero of 10

Find and graph (on common axes) the partial sums up to that S"'o whose graph practically coincides with that of/ex) within graphical accuracy. State what 1Il0 is. On what does the size of IIlO seem to depend?f(x) = 11ta1,1x), where a1,1 is the first positive zero of 11

CAS EXPERIMENT. Fourier-Bessel Series. Use Example 3 and again take n = 10 and R = 1. so that you get the series (19) flx) = al10lao,lX) + a210(aO,2x) + a310lao,3x) + ... with the zeros 0:0,1 0:0,2' .. from your CAS (see also Table AI in App. 5). (a) Graph the terms 10(aO,lx), ... , 10(aO,lOx) for o ~ x ~ l on common axes. (b) Write a program for calculating partial sums of (9). Find out for what f(x) your CAS can evaluate the integrals. Take two such f(x) and comment empirically on the speed of convergence by observing the decrease of the coefficients. {c) Take f(x) = 1 in (19) and evaluate the integrals for the coefficients analytically by (24a), Sec. 5.5, with JJ = I. Graph the first few partial sums on common axes.

TEAM PROJECT. Orthogonality on the Entire Real Axis. Hermite Polynomials.13 These orthogonal polynomials are defined by HeoO) = 1 and REMARK. As is true for many special functions, the literature contains more than one notation, and one sometimes defines as Hermite polynomials the functions dn _X2 2 e H *(x) = (-l)ne" --- n dxn This differs from our definition, which is preferred in applications. (a) Small Values of n. Show that He1(X) = x, He3(X) = x 3 - 3x, He2(X) = x 2 - 1, He4(X) = X4 - 6x2 + 3. (b) Generating Function. A generating function of the Hermite polynomials is (20) -n etx-t2/2 = L anlx)fn n=O because He.,(x) = n!anlx), Prove this. Hint: Use the formula for the coefficients of a Maclaurin series and note that tx - ~f2 = ~X2 - ~(x - t)2. (C) Derivative. Differentiating the generating function with respect to x, show that (d) Orthogonality on the x-Axis needs a weight function that goes to zero sufficiently fast as x ---? ::,:::cc. (Why?) Show that the Hermite polynomials are orthogonal on -<Xl < X < cc with respect to the weight function rex) = e-x2/2. Hint. Use integration by parts and (21). (e) ODEs. Show that (22) He~(x) = xHen(x) - Hen+l(x). Using this with n - 1 instead of nand (21), show that y = Hen(x) satisfies the ODE (23) y" - xy' + ny = O. Show that w = e-X2/4y is a solution of Weber's equation14 (24) w" + (n + ~ - !x2 )w = 0 (n=O,l,).

WRITING PROJECT. Orthogonality. Write a short report (2-3 pages) about the most important ideas and facts related to orthogonality and orthogonal series and their applications.

What is a power series? Can it contain negative or fractional powers? How would you test for convergence?

Why could we use the power series method for Legendre's equation but needed the Frobenius method for Bessel's equation?

Why did we introduce twO kinds of Bessel functions, J and Y?

What is the hypergeometric equation and why did Gauss introduce it?

List the three cases of the Frobenius method, giving examples of your own.

What is the difference between an initial value problem and a boundary value problem?

What does orthogonality of functions mean and how is it used in series expansions? Give examples.

What is the Sturm-Liouville theory and its practical importance?

What do you remember about the orthogonality of the Legendre polynomials? Of Bessel functions?

What is completeness of orthogonal sets? Why is it important?

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) y" - 9y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) (1 - X)2y" + (1 - x)y' - 3y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.)xy" - (x + l)y' + y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) x 2 y" - 3xy' + 4y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.)y" + 4xy' + (4x2 + 2)y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.)x 2y" - 4xy' + (x 2 + 6)y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.)xy" + (2x + I)Y' + (x + l)y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) (x 2 - l)y" - 2xy' + 2y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) (x 2 - l)y" + 4xy' + 2y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) x 2y" + xy' + (4x4 - 1)y = 0

Find a general solution in terms of Bessel functions. (Use the indicated transformations and show the details.)x 2y" + xy' ., (36x 2 - 2))" = 0 (6x = z)

Find a general solution in terms of Bessel functions. (Use the indicated transformations and show the details.)x 2y" + 5xy' + (x 2 - 12)y = 0 (y = u/x2)

Find a general solution in terms of Bessel functions. (Use the indicated transformations and show the details.)x 2y" + xy' + 4(x 4 - l)y = 0 (x 2 = z)

Find a general solution in terms of Bessel functions. (Use the indicated transformations and show the details.)4x2y" - 20xy' + (4x 2 + 35)y = 0 (y = x 3 u)

Find a general solution in terms of Bessel functions. (Use the indicated transformations and show the details.)y" + k 2 x 2y = 0 (y = uVx, ~kX2 = z)

Find the eigenvalues and eigenfunctions.

Find the eigenvalues and eigenfunctions.

Find the eigenvalues and eigenfunctions.

Find the eigenvalues and eigenfunctions.

Find the eigenvalues and eigenfunctions.

Write a program, develop in a Fourier-Legendre series, and graph the first five partial sums on common axes, together with the given function. Comment on accuracy. e2x ( - 1 ~ x ~ I)

Write a program, develop in a Fourier-Legendre series, and graph the first five partial sums on common axes, together with the given function. Comment on accuracy.sin ( 7TX 2) ( - 1 ~ x ~ I)

Write a program, develop in a Fourier-Legendre series, and graph the first five partial sums on common axes, together with the given function. Comment on accuracy. 11(1 + Ixl) (-1 ~ x ~ 1)

Write a program, develop in a Fourier-Legendre series, and graph the first five partial sums on common axes, together with the given function. Comment on accuracy. Icos 7Txl (-1 ~ x ~ 1)

Write a program, develop in a Fourier-Legendre series, and graph the first five partial sums on common axes, together with the given function. Comment on accuracy. x if 0 ~ x ~ 1,0 if -1 ~ x < 0