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Unformatted text preview: Name: Section No. If you want your grade to be posted {with the last four digits of your Social Security Number) on the course web page, check here: El 110.302 Differential Equations December 14, 1998
Final Exam One sheet (2 sides, 8% x 11) of notes may be used during this exam. A table of
integrals is also provided. No books, calculators or other materials are permitted. This exam contains 9 pages and has a total of 200 points.
Time: 3 hours Please put your name on the top of each page. Part 1. Multiple Choice (60 points) Circle the letter corresponding to the best answer to each of the following problems. Each correct answer
is worth 6 points and no partial credit will be given. You need not show your work on the multiple
choice problems. A differentiable function f (:5), 0 S a: g 1, with f ( 1) = 0 has a Bessel series expansion W) = cam) , where J0 is the Bessel function of the ﬁrst kind of order 0 and
0<C1<42<§3<...<Cn<... are the zeros of Jo on the positive :E—axis. A B C D @ F 1. Which of the following gives the formula for the coefﬁcients on of
this expansion? A. cn = fol f(1:)$”da:
1
B. cn =/O f($)cosn7m: da:
C. 0n = /O1 sin mm da:
D. em = [01 f<r)Jo(<nx)dr
E. on = /01 mf(a:)J0(Cna:)dx //01:I:J0((n$)2dr
F. cn = /01 :cf(:c)J0(Cn:L‘)d:E/A1:Ef(aj)J1(an)d:E " Vaﬁj/‘gq / Name: page 2.).
Which of the following statements best describes each of the following differential equations?
A B C D E [email protected] 2. $2y”+3:y’+(g;2_9)y=0
[email protected] D E F G 3. $2y"+3y’+2y=0
All B C D E F G 4. (cosz)y”+:ry’+2y=0
A B @D E F G 5. 21:2y”+(534+x)y’+(2$—3)y=0
A The point a: = 0 is an ordinary point.
B The point 2: = O is an irregular singular point.
C. It has a solution of the form 1’1 + a1 + (1223 + (13:32 + . . ..
D. It has a solution of the form z'l/z + alarm + (12:1:3/2 + (1315/2 + . . ..
E It has a solution of the form 101/2 + (119:3/2 + (1225/2 + a3x7/2 + . . ..
F It has a solution of the form I2 + (113:3 + (121:4 + a3x5 + . . ..
G None of the above.
Which picture describes the type of each of the following critical points?
E = 2:1: —— 3y
A B @ D E F G H I J 6. the critical point (0,0) of { g; =7z_2y
49“ = r —— my
A B C D E®G H I J 7 thecriticalpoint(e1)of{:;
dt = x — yey
. dz _
a; — y
@ B o D E F G H I J 8 the critical point (0,0) of d _
_ if = —9Asmx — 4y
A B C D E F G @ I J 9 the critical point (7r, 0) of the system in #8 above d2:
_ I I E =x(1.5—:L‘—O.5y)
A B C 'D E F G @ I J 10. the critical pornt (1,1) of{ in = y(2 _ 0.5?! _ 1.5x)
sinlr source  sink source
A B I C D E sink source sink source 30LU7/ﬂ/I/j' Vale"?ch (2 Ifyou want your grade to be posted (with the last four digits of your Social Security Name: Section No. 60 Number) on the course web page, Check here: D 110.302 Differential Equations December 14, 1998
Final Exam One sheet (2 sides, 8% X 11) of notes may be used during this exam. A table of
integrals is also provided. No books, calculators or other materials are permitted. This exam contains 9 pages and has a total of 200 points.
Time: 3 hours Please put your name on the top of each page. Part I. Multiple Choice (60 points) Circle the letter corresponding to the best answer to each of the following problems. Each correct answer
is worth 6 points and no partial credit will be given. You need not show your work on the multiple
choice problems. A differentiable function f (3:), 0 S a: S 1, with f (1) = 0 has a Bessel series expansion f($) = imam) , where J0 is the Bessel function of the ﬁrst kind of order 0 and
0<§1<C2<43<...<§n<... are the zeros of Jo on the positive x—axis. A C D E F 1. Which of the following gives the formula for the coefﬁcients en of
this expansion? 1 A. on = $f($)Jo(<nx)d$ //Olrcf(’C)J1(Cnx)dx 0
l B. en = xf(:t)J0(Cna:)dct //01 xJ0(Cn:t)2dx 0
1 C. en = f(:t)$"d1;
0
1 D. on = f(r) cos mm: dz
0 1
E. en = f(x) sin mm: dx
0 1 F. Cn: D f($)Jo(Cn$)d$ Name: L V a Mary/5M 2‘ page 20 Which of the following statements best describes each of the following differential equations? A B c D E F G 2. 2:2y”+3y’+2y=0
rs;
A B C D®F G 3. (cosx)y”+1:y’+2y=0
(Al B c D E F G 4. 2x2y”+(5x4+x)y’+(2m—3)y=0 A B c D E [email protected] 5. mzy”+:cy’+(:z;2—9)y=0 A It has a solution of the form :0‘1 + a1+ agx + 0,3302 + . . .. B It has a solution of the form 271/2 + (1129/2 + agrm + 0315/2 + . . ..
C. It has a solution of the form 021/2 + (1113/2 + a2x5/2 + a3x7/2 + . . ..
D. It has a solution of the form 1:2 + £11223 + (123:4 + 413235 + . . .. E The point 2: = O is an ordinary point. F The point :5 = 0 is an irregular singular point. G None of the above. Which picture describes the type of each of the following critical points? do:
ch _ = y
@ B c D E F G H I J 6. the critical point (00) of j‘ .
3% = —9 8111117 — 4y
A B C D E F G (Iii) I J 7. the critical point (7r,0) of the system in #6 above “'“‘ 2—"; = 22: — By
A B 1gb E F G H I J 8. the critical point (0,0) of Q _7$_2
dt " y
5‘5 = I — my
A B C D [email protected] H I J 9. thecriticalpoint (e,1)of{:;$_yey
dt " (A) . _ ' f % = z(1.5  x — 0.5g)
A B C D E F G I J 10. the critical pomt (1,1) 0 ﬂ = y(2 _ 0.5?! _ 1.53) sink source ' sink source
B t C D E
sink source sink ' source Name: l y page 3 Part 11. Express all answers using real numbers and real functions only. (You may use complex num—
bers in your work, but \/—1 should not appear in your answer.) To receive full credit you must show
all of your work and write your answers in the indicated places. 1. Find the general solutions of the following differential equations: a) (15 points) — 2 Answer: /”‘ SIXvﬂtu/X/fc b) (15 points) (sec2x+3m2tany—:—f) d$+ (mllsec2y+4y3+:i:) dy=0
WA, W
7, al/l/
w: 711W1/~ {Z— _: a t/ N M y
L
n W W e + w
’b
3.5,“ : A/J {heel/w [fﬂ‘gli
3 V 7.
: f3MZ/+% ¢ A A 7M: é/e/J' A/u/xy: Answer: 5m, VT/a/z/j‘ Name: page4
2. a.) (15 points) Find the general solution of the differential equation
ty”—(t+2)y’+2y=0.
Hint: One solution isy=e‘. 1 I/ 161‘} /A W" V”?
f ‘Vef V]: ’26" P :6 6
y! :8 V1 « , y, // C
{ r
5FJ%:5(/*%)stf*ij’° l f
7— _.
V/: —2f8{fQA/f/:e~2feézf : 1L 6
_ z r 4‘ v4
V:jf’ze{/Z;;,/e +j2fe c/a
, {13425124461
v: é’fz’ f7; 22“"1)
V1 3 " 1‘1 5'1
y; 6/ y/ {« CL (AL) 5 814+ Cl (211742751‘2) ___/_ Answer: y = r3: ,2; =
Name: C” " /" Ivy/.5 2. b) (15 points) Find the general solution of the differential equation page 5 Answer: 7 f 1‘ f f 92 Name: 1' 5) page 6 3. a) (10 points) Give a second order linear ODE whose solution is y = Clet cos 375 + 026t sin 3t . // / 3" _.
Answer: ﬂwlkO—ﬁ b) (10 points) Find the Wronskian of two (diﬁerent) solutions y1,y2 of
(1 — t2)y" — 2153/ + 6y = 0. (Your answer should have an arbitrary constant 0. Do not try to solve the differential equation.) / Zz" 6'
ya 75%: y/f/ZEJ/‘ﬂ
u; 23— ; Z}:
‘ [vz‘t 514/
W:Cejpa/f
2r  /z’2/
WM /
 H“ ._._
W: (8 A 5 t‘t/ 2.
Answer: W = t '/ Name: page 7
3. C) (10 points) Find a second order linear ODE Whose solution is L y=01t+c2ef . W: )7: :2 ;’ {294/61,
Zn W: * j/Uc/f r ﬁr f
’0 31%“ ’ ‘ jive 2“ 77 WWI/77" z‘
ny 37 0+ {Hg/{:0 37,9" g F649;? 8'ff/7(/e*7fcag’f—‘0
/’/7%V70 57 /”L5"5:0
c L ii
3:/%*/ / p‘ 257/ AnswetMX//+Zly/_1/ 5 0 Name: L y 75‘ page 8 . (25 points) Suppose that a metal rod 20 cm long with uninsulated ends and thermal constant
a2 = 4 initially has a uniform temperature of 0°C throughout. At the initial time t z: 0, the right
end (30 = 20) is instantaneously heated to 100°C and kept at this temperature, while the left end
(:1: = 0) is kept at 00C. The temperature distribution u(.r, t) of the rod is governed by the following
partial differential equation, boundary values and initial condition: (92u Bu
4% = a, u(0,t) = 0, u(20,t) = 100, u(.r,0) : 0. Find the temperature distribution u(:(:, t). Hint: As if —> +00, u(:c,t) approaches the steady—state temperature 2 5.10. Let f(:1:,t) =
u(;1:, t) — Write the PDE, boundary values, and initial condition for f and solve. m] a: M 0; 2V D 93.53 : Q20 Marja/0 raw/:0
317' m4 ’
79(60J‘r57‘ ,171'7714‘75 my
I 3 WT ‘ ————)(
ferf/ Etna lit M220
—ﬂ 14/7“
:Zéﬁg wafﬂe/ti}!
1r)
10 h ﬁr
77’  J. “’[lzf
é : 42’ S (efﬂmagl/Xe—LS/sza
’7 Q‘O [2 a
L n77" 2y 2 ’1’I,r,//—Z£ rare/23k 7‘07}? M10
SIM?!) " M77“
Lii(—Z§ Mom/1777) *0
I h“ 1 ’7 Zoo
14 if
r 14 .20.[—/] :[4/ ,7”.
‘ 1777‘
00 _"‘L”1.
Answer: u(a:,t)=51.+Z (_,}”%€8 W‘éMg/r
n2] 50L UT/ﬂﬂ/Y Name: page 9
5. (25 points) Find the solution of the initial value problem: due d—tl : £131 — (1)2 , Z 5 d]; d—: = 4331 — 3$2 , = 7
Hint: The characteristic equation is r2 + 21" + 1 = 0. I I I ‘ [/ 1g/ ‘f
, f r ; 71b:  / It, /; [ 
4—L#~3 I £.8 /;Qé—é
2226' AI—‘ﬂ/ éa7’ ‘30:) ‘3/1/
ﬁrm/d {@173 4*] 1:6/ fC'L/f
Lax'f/ﬁ
' :X[g : C
y:(CPCJf/€f I 6’ /)_l C
’ ( lab/Ne” 77: 11/0)“ 30/“ 2
*1‘[:QC/*CQ( 5/0’51
C;:/0'7:?
17w («Late/j; manta/j: /W6£«3 :77‘Kf 1:1(t) = 6 if
Answer: { 3320:) Z (7% «2" Note: Your answer should consist of real functions only and should not contain vectors. ...
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 BROWN
 Equations

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