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Unformatted text preview: AM 351 Midterm Examination — Fall 2003 BESSEL FUNCTIONS FORMULA SHEET Bessel ’5 DE: 22y” + 211’ + (I2  ﬂy = 0 Bessel ’5 DE in normal form: 1—417
u”+ (1+ 412 2 )u=0, with y: u. i
ﬁ DES transformable into Bessel ’5 DE: 1) 12y" + (1 + 20)Iy’ + WI” + c”  b2P2)y = 0, w = 1‘11, 2 = ﬁzb.
ii) xzy”+ (1+ MW + [(a2 + 8):” + car — p’Jy = 0,
w = any, 2 = :12. Bessel function of the ﬁrst kind: ' ' __ :5 P °° (—1)" x 2!:
JP“) ‘ (E) gk!l‘(k+p+ 1) (E) '
forallp#——l,—2,..... ’ Asymptotic form as x ~—> 0”: JP“) = my: 1) [1 ‘ 17% (92+ ‘
for 21le $5 —1,—2,... m2) ~ %In (—3), Y,.(z:)~—%(n—1)! ", n=1,2;.... Asymptotic farm as :r —r +00: Jp(z) z gees (z — (212+ 1H), Yp(:c) z ﬁsh: (z — (2p + 1H) , forallp#—1,—2... FORMULA SHEET (cont’d) Identities: Zp+:(=c) = 32pm — 2,265) Zp_1(z) = 22142:) + z;(x) I
Zp+1<z)+ z 1($)= gfzpm Where Z1, stands for J}7 or Y1}. The gamma function Hz): Transformation of a Second Order Homogeneous ODE to “Normal Form” Given the homogeneous DE
y" + P(z)y’ + Q($)y = 0,
the substitution I
mm) = eif”<=)d“u(z)
yields the following DE,
u" + q(z)u = 0.
where (1(3) = (2(2) — Emmi “124313). 1. Consider the following class of linear second—order DEs,
y”+P(z)y’+Q(w)y=0, (1)
R where P(:c) and Q(z) are assumed continuous over an interval I C R. (a) Let u1(:r) and u;(z) be two solutions to the DE in Eq. (1). It is said that the condition .—_/ of “linear independence” must be satisﬁed by these two solutions so that. so that a general
' solution 310(1) can be constructed from them. Deﬁne linear independence mathematically ( V and also the mathematical condition — call it “MC” — that guarantees it. Provided that u this condition is satisﬁed, what will then be the form of the general solution go in terms of u] and ug? (b) What does the statement, “ya is the general solution to Eq. (1)” mean? More mathemati
cally, show why the condition “MC” in (a) must be satisﬁed for yg, as given in terms of u]
and 112, to be the general solution to Eq. (c) Explain why you need to check that the “MC” condition in (a) is satisﬁed only at a point
In E I as opposed to checking that it is satisfied over the interval I. You should refer to a
theorem here — but you do not need to prove the theorem. (d) Suppose that the two solutions 114(2) and u2(:c) do not satisfy the “MC” condition in (a).
What can you do to come up with another solution to Eq. (1)? Just state the method —
you do not have to derive any DEs or solve anything here. If your method is successful,
can you now construct a general solution? Justify your answer. 64*in Md M10!) We linian iWPm/JM ;[§/UJ(
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do, ﬁctions [A (X) 0'», . J
W soumm; ' M Mr) . i MLC may M 9&1sz . Mn M1(6{)U‘1’(5{)_ M‘/(’x) MLLY): O ~ Q N I l  o ' l » I
/ EN «0 A M, Del’gﬁ m4~ (2351' M‘ (Al/(X)+ M‘t’x) Ml (OX) —u,'(«) MLY’X} “ U."(’>()M1l”X)= O
1’) U.(7<)MLK‘(’X) ~ /
U! (L 1W”! (A’J'mlkm
/
\d i WHi’Xl Ulzi‘x) = O 2. Consider the second order DE \ i I; :
my" + iy’ + my = 0. (2) / ng) Using an appropriate power series method, ﬁnd two solutions y1(z),y2($) of this DE that
could comprise its general solution. Brieﬂy justify the choice of power series method used. C A complete solution should include the recursion relations for the series coeﬁicients a". You
/ need only write each series solution ~ apart from its multiplicative constant ~ out to two
terms. You do not have to determine a closed»form expression for the coefﬁcients an in 7/ ‘ terms of the ﬁrst coefﬁcient, e.g. (10. What is the dominant (as opposed to the limiting) behaviour of each of these two solutions as 1: —) 0+7 ‘ (b) Determine the normal form DE
u" + q(z)u : 0 (3) asSOciated with the DB in Eq. (2). Use your result to determine the behaviour of zeros of
Solutions y(z) to Eq. (2) on the positive real line. i.e. ﬁnite number of zeros vs. inﬁnite
number of zeros. In your solution you should make reference to any appropriate theorem(s)
on qualitative properties of DES. ' (c) What, if anything, can you say about the relationship between zeros of the solutions y1(r)
and 342(1) found in 2(a)? If anything can be said, state the name(s) of the appropriate
theorem(s) used. (You don't have to state the entire theorem(s).) (d) me Eq. (3), write down the asymptotic behaviour of solutions y(:t) oqu. (2) as I —> +00. (e) With reference to another appropriate theorem on qualitative behaviour of solutions, corn»
ment on the spacing of consecutive zeros 52min“, of any solution to Eqs. (2) or (3). For
example: Does the spacing approach a limit a? If so, does it approach the limit from above or below? We Will We ‘m wihoﬁ oi Rbtanm sinus. 'llnr D€ ton (Wu, @4 5 1 \IV ’ a
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2’. he I" D
IXU: = C a Yner¢z ,. / '
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7:!) .l
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(taxman (n+2) (Ml) em atio‘n — Fall 2003 3. This question continues With an examination of the DB in Eq. ,(2), (a) Show that the general solution to Eq. (2) can be expressed in terms of Bessel functions, (b) a“.
‘1, using the “formula sheet." Then match each of these Bessel function solutions with the
power series solutions y; and y; from Question 2(a). What is the dominant (as opposed to
the limiting) behaviour of each of these two solutions as x —> 0+? From the asymptotic behaviour of Bessel functions as I —> +00, show brieﬂy that your
results in 3(a) are consistent with the result of Question 2(d). Suppose that the DE in Eq. (2) ariSes from the study of a hanging chain'type vibrational
problem as was done in class. Recall that the position u(1,t) of a mass element at a point
0 g I S I and time t 2 0 is assumed to have the form u(x,t) : 10(1) coswt. Now assume that in this problem the amplitude function is given by : y(3w7§), where y is the solution to Eq. (1) in terms of Bessel functions. (You do not have to derive any wave
equations here or perform any analysis of the physics  simply read on.) Show that there exist an inﬁnite number of solutions 10,1(1), n = 1,2, . .. to this problem
satifying the conditions 10(0) = A, 112(1) :2 0, where A and lare positive real constants. Give
the frequencies of vibration w" in terms of appropriate quantities that are characteristic of
the Bessel function(s) used in the solution. (You do not have to know the numerical values
of these quantities.) Using your results, sketch the qualitative behaviour of the ﬁrst three
such vibrational modes (assume them to be for a hanging chain). Determine the positions of any nodes in [0,1] for the three vibrational modes. ma max; wrilﬁ “ll/Q DE in Perl @ 06 «2 + CHIC/alle + («1+ ("all("01> : O
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when we Aetem‘wci zr: matmn—Fajr . > 12 ' 4. State and prove the Shim) comparison theorem. Shaw Comgmicon harem:
[1% UR} ﬁnd uh} \ae eruH soluﬁons +3 44M DES \
M00 + (300‘ ‘(x} =0 and
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 Spring '08
 SivabalSivaloganathan
 general solution, Normal Form, Bessel function, Bessel Functions, power series method, Bessel Function Solutions

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