×

### Let's log you in.

or

Don't have a StudySoup account? Create one here!

×

or

18

0

22

# Class Note for MATH 3331 with Professor He at UH

Marketplace > University of Houston > Class Note for MATH 3331 with Professor He at UH

No professor available

These notes were just uploaded, and will be ready to view shortly.

Either way, we'll remind you when they're ready :)

Get a free preview of these Notes, just enter your email below.

×
Unlock Preview

COURSE
PROF.
No professor available
TYPE
Class Notes
PAGES
22
WORDS
KARMA
25 ?

## Popular in Department

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

×

×

### What is Karma?

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

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

×

×

### BOOM! Enjoy Your Free Notes!

×

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

Steve Martinelli UC Los Angeles

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

Allison Fischer University of Alabama

#### "I signed up to be an Elite Notetaker with 2 of my sorority sisters this semester. We just posted our notes weekly and were each making over \$600 per month. I LOVE StudySoup!"

Steve Martinelli UC Los Angeles

Forbes

#### "Their 'Elite Notetakers' are making over \$1,200/month in sales by creating high quality content that helps their classmates in a time of need."

Become an Elite Notetaker and start selling your notes online!
×

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com