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# Class Note for MATH 3331 with Professor He at UH

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This 22 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 18 views.

## Reviews for Class Note for MATH 3331 with Professor He at UH

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Date Created: 02/06/15

Lecture 29 9 2 Phnzv 5mm 9 3 pm Pbquot mm x Wm H Wm M WM WW M mm quotA divlavamZ mm Section 92 7 2D Distinct Eienvalues In Class Exercises Planar Systems A tK Pm 09115 1 szol 12 a b Set EzaI d traceofA 2d Systems A Sec 924 393 C d QZCLd bc detA gt pgt2 TD pg zdetqa b 0 Roots Ofp2 C d O A A172Ti T2 4g2 a A b C A g u a Ad A bc Roots are real and distinct if 2 adad b5gt T2 4D gto 4 Mai IIll Jiwen He University of Houston Math 3331 Section 19470 Lecture 29 April 15 2009 2 22 Section 92 2D Distinct Eienvalues ln Class Exercises Solutions of 2d Systems for Distinct Real Eigenvalues A a b E aI d c d D ad bc 7 2 7p was Assume T2 4D gt o g 39 gt A has two distinct real qe i genvalues A172 X05 X1 073905 00 quot 1 quotGeneral Solutionzn Let v 0 be in nuA A11 xtc1X1t X205 2 75f3 32720 be in nuA 2I quot v1v2 are linearly independent vw39ll 2CI t 11 ij s a gt Fundamental Solution Set X1tffV17 X205 eAQtV2 A all A A7 I T fraying yKGWVLalW sold39w ll Fundamental Matrix Jiwen He University of Houston Math 3331 Section 19470 Lecture 29 April 15 2009 3 22 Section 92 L 31 2D Distinct Eienvalues In Class Exercises Example V WA 439 11quot Ex A 4 gt p A2 A 2 A 2gt 1 gt mnvalues A1 3 A2 1 46 35 T1D 2 1 14 0397 1 Fundamental matrix Ida 3i LE 3 i 7ltA2Igt i 8 i Xt i 2861 MI i 3 2 i Hamil i 8 i 2ZTt iget i X096 ei envectors 1 1 1T 9 V2 271T x1lttgt 5iiix2lttgt5iii are a funda ental set of solutions Jiwen He University of Houston Math 3331 Section 19470 Lecture 29 April 15 2009 4 22 Section 92 2D Distinct Eienvalues Iii Class Exercises Exercise 923 j alas Ex 92mm solution of y Ay for A 5 1 1 My fur 7A r2 A yaw A111 C39s quot f 4 T 7 D 12 x T2 4D 1 gt eigenvalues A12 72i 12 x I Lf 1 3 2 4 Find eigenvectors I S 2 1 1 1 d7 2 1 zlw lll gt Fundamental set of solu 39 s 7 yzaiwiii h 270 I 39 General solution 1 a V t r 39 v u G 39I i t i t yt C1Y1t C 3 2 268 7 70 Jiwen He University of Houston Math 3331 Section 19470 Le ture 29 April 15 2009 5 22 Section 92 Exercise 929 39Ex 929 Find solution of system of Ex 3 for IC y0 O 1T h 976 E21 m J Match 017C2 to IC ylt0gti ii iaHr en iii 2H ii HEi i i Mi i 61 390 z 0 2 7 C12 quot1 La 39r f 1 19 Liz 1 1111 April 15 2009 6 22 Math 3331 Section 19470 Lecture 29 Jiwen He University of Houston Ci AS Czhh iCase A T2 4DgtO a4 20 fJ ak gt real distinct eigenvalues ab A12Tmz2 XQ39 Cit 21413 db Case T2 40 lt 0 gt complegt lt eigenvalues A172 aii Tad Dad bc 7 a T2 4D T22 p2 T D Lj N s CasegT2 4D20 T T gt single eigenvalue A T2 wtkhiisf wi 39 zi Iii Jiwen He University of Houston Math 3331 Section 19470 Lecture 29 April 15 2009 7 22 quot Section 93 Distinct Eienvalues Comlex Eienvalues Case A T2 4DgtO Case A T2 4D gt 0 gt real distinct eigenvalues A192 2 TlL T2 4D2 o The 4 half line trajectories separate 06 4 regions of R2 l V General Solution v1 V2 eigenvectors Xt cleAltVJ l 02 A2E y Full lines generated by v12 22 Trim Wt i Half line trajectories W 0 if 02 0 gt Xt C A175V1 gt trajectory is half line H1XC V1iCMgtOif01gtO 3Xavlialt0ifcllt 0 Same fochITO 02gtO orltO Jiwen He University of Houston Math 3331 Section 19470 Lecture 29 April 15 2009 8 22 all Section 93 Phase Portrait Phase portrait Sketch trajectories Indicate direction of motion by wooint ing in thew Direction of Motion on Half Line Trajectories o If A1 gt 0 then 105 2 cleAltvl moves out to 00 for gt oo 70 outwards arrow on HHS approaches 0 for t gt oo If A1 lt 0 then xt aleaim apUF oaches 0 for t gt oo inwards arrow on HH moves out to 00 for t a oo Jiwen He University of Houston Math 3331 Section 19470 Lecture 29 April 15 2009 9 22 Section 93 Saddle A1 gt 0 IgtNA2 h k Saddle gt315 A Igt O gt A2 I HaIaJine trajectories Generic Trajectories L2 t a V V v 149 y 7 I 9 X VI e snu my 9 WMquot k i 4 7 L Generic trajectory in 2 ea 39 pro T 0L1 10er 2abwl 0L2 forte co Jiwen He University of Houston Math 3331 Section 19470 Lecture 29 April 15 2009 10 22 r a Section 93 Nodal Source A1 gt A2 gt 0 v 39a Vl Nodal source JAPAN Half line trajectories Generic Trajectories l392 y quoti TM 9 N70 7 v ast L 1 f gdv a x fast escape to 00 39 2 603 W IS WV 65 0 parallel to L1 for 4 4a A I V taoo o tangent to L2 for fut 639 H t gt OO Jiwen He University of Houston Math 3331 Section 19470 Lecture 29 April 15 2009 11 22 Section 93 Distinct Eienvalues Comlex Eienvalues Nodal Sink A1 lt A2 0 39 39 7 L VI 4 Nodal sink A1 lt A2 lt 0 Half line trajectories Generic Trajec Dries L quot E 2 y AK 0 39 fast 4 A slow i 39 ALlt D L L 2 gt fast approach to 0 Generic trajectory is 0 parallel to L1 for t a oo o tangent to L2 for t a 00 H Jiwen He University of Houston Math 3331 Section 19470 Lecture 29 April 15 2009 12 22 Section 93 Saddle Example Saddle gt 1 4 Ex A 2 1 H54 L1ltgt A123 HV121T 1 T H A2 3 V2 1 1 Time Plots for thick trajectory 30km 39 x x4y y 2X y y zw i N gt x i E N E 2 U i J N N xi 9 a 2 V g 1 L 3 N N 2 7 gt 0 Home X J x v Z Z k y K T i T f 7 6 K K V V e e S quotN r 39 39 e e lt amp K R w ewewlt lt lt lt ltm s m ltm g i l 5 6 xv 5 Math 3331 Section 19470 Lecture 29 April 15 2009 13 22 Jiwen He University of Houston Section 93 Nodal Source Example Nodal Source v M Ex y Lt 5i Y 39l N Y YT T277 77739 739 J QR xxx T377 777739 27 Kgxx wa rgrr 7 71157 15 gsxx R Y T rgr 777 22715 gt RKK 39 39V TE7 7 7 739 7277 I Flt ampRV I TE 77 21239 271 U l eee K 37 739 222239 2 C l eekelt 7 722 510 gt k 2 gtlt zzzz 99 aquot jzzzzzz 1 N zitKKK z x gtgt gt gt 5 KitK lacz lIll 15 Z c I 1422 Math 3331 Section 19470 Lecture 29 April 15 2009 Jiwen He University of Houston Section 93 Distinct Eienvalues Comlex Eienvalues Nodal Sink Example Nodal Sink 39 3 1 Lu Ex A 1 3 I A1 4 lt gt v1 11T w 9 I x 3X y X By L 20 X 5quotquot5 quot39 39X quotquotampquot39 quotquot 39JJ 39Zquot39ZquotZquotZquotquot39 quot y N sxttt 5 xzzzzz E N N N x t M 2 Z 1 z M slI l 3 NL g XXLiglE 15 s 5amp8 N x x t 52 2 z z z z zz gt ggt8x x 5 z t zzgm U gtgt N s i z x 11sz N z iLAZKKKE C 9 9 quotgt gt 13 KKZLL U s s H 22272 2 gt gt K K 1 gt 0 I u UK x 2722 2 239 2 73 R 22 22239 7 751 s 222 Z 739 739 7 T T z 2 7 T T f f f 7 7 7 7 T T T 22 739 f 7 7 7 T T 1 7 f F f 7 f f 7 T 7 F 7 7 7 7 7 7 7 5 1 5 o gtlt Jiwen He University of Houston Math 3331 Section 19470 Lecture 29 April 15 2009 15 22 K lt cigarA wltu 1 a m mm Wm I em r W W mcmwcme rm W 2 mquot anma u my 5m same amp gtu ee gnvaimvulwzumpblt 5m Smk u m zw mmm Dicumr emmngcx r cmqazu my ham svv some a gtu 0 5altwsm we mm a a 7 Qusmmws WA 3 q H Section 93 Center 04 O Center 04 0 gt xt periodic X gt trajectories are closed curves Direction of Rotation At x 10T y c If c gt 0 gt counterclockwise If c lt 0 gt clockwise 1H Jiwen He University of Houston Math 3331 Section 19470 Lecture 29 April 15 2009 17 22 39 Section 93 Spiral Source oz gt 0 Spiral Sourceagt0 gt growing oscillations X gt trajectories are outgoing spirals Direction of Rotation At x 1 OT y c If c gt 0 gt counterclockwise If c lt 0 gt clockwise I Jiwen He University of Houston Math 3331 Section 19470 Lecture 29 April 15 2009 18 22 Section 93 Spiral Sink oz lt Spiral Sink oz lt 0 gt decaying oscillations gt trajectories are ag 0 ingoing spirals Direction of Rotation At x 1OT y c If c gt 0 gt counterclockwise If c lt 0 gt clockwise H Jiwen He University of Houston Math 3331 Section 19470 Lecture 29 April 15 2009 19 22 Center Example Center 10 4 Ex A 4 2 A2ilt gtV K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K Jiwen He University of Houston K K 2 K x 4x10y y 2x4y K Section 93 N 1 Time Plots for thick trajectory Math 3331 Section 19470 Lecture 29 April 15 2009 20 2 2 Distinct Eienvalues Comlex Eienvalues Section 93 Spiral Source Example Spiral Source 02 1 ExA 1 0392 1 A 02 z lt gt V 7 Time Plots for thick trajectory K it KKK i L L RR 99 KZ RRR RRR e Math 3331 Section 19470 Lecture 29 Aril 15 2009 21 Jiwen He University of Houston Section 93 Spiral Sink Example Spiral Sink O2 1 Ex A 1 O2 A O2 I z lt gtv Time Plots for thick trajectory I i a x x I 3N 3amp 22 A A1 gt 75 1 L C f39 E L U gt s z a a X T f 3 J L i V Z J V I Y R R R 7 K K a Z Z x ix x x R 6 K 2 z z z z 05 V V i R S z SR KKltSltHKKKEMIZZZIZ R quot R N 9 e k l Z Z Z ampltelt kg 1 05 0 05 1 Math 3331 Section 19470 Lecture 29 April 15 2009 22 22 Jiwen He University of Houston

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