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Class Note for MATH 3331 with Professor He at UH

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This 12 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 35 views.

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Date Created: 02/06/15
Lecture 3 521an 2 7 Exmence and Unmuenm n1 swam 1W He Wm M WM WW M mm m KVLuNnvmmzn Section 27 v r Existence and Uniqueness Theorem Basic Existence and Uniqueness Theorem EUT Suppose ftz is defined and continuous and has a continuous partial derivative 8fta8x on a rectangle R in the tar plane Then given any initial point 260900 in R the initial value problem 22 ux 20050 930 has a unique solution 9305 defined in an interval containing t0 Furthermore the solution will be defined at least until the solution leaves R Jiwen He University of Houston Math 3331 Section 19470 Lecture 3 February 639 2009 2 12 Section 27 Example EX tx ZxI 3t2 m chtI St o f and 8f8zv are defined and continuous for any ta if t at 0 o General solution use Sec 26 45y 9325 2 3752 Ct o For any 0 320 2 0 hence no solution for 320 2 x0 73 0 A I quot01 wt 00 solutions for 920 2 0 2 2 0 Solution for xto 2130 to gt 0 3t CtOxo 0270750 3750 4 v 3 05 3752 550750 3150 Figure 1 All solutions of 72 unique solution with IoE 000 paSSthrough 0 0 EUT applies to any rectangle that is not intersected by the ver Ill tical line 25 0 Jiwen He University of Houston Math 3331 Section 19470 Lecture 8 February 6 2009 3 12 Section 27 1 v Example Non Uniqueness of Solution r 06 yt Ex 12 9013 SOV fail3cm 32C23 tl D gt BiOf i23ti C32 C 2D3 0 Let C O gt i0 O 02quot o Other solution with MO 2 0 3315 O 7 X gt At least 3 solutions for IC 0396quot 5170 2 O o EUT doesn t apply to any rec tangle that is intersected by the up horizontal line 1 0 1391 1quotZ Jiwen He University of Houston Math 3331 Section 19470 Lecture 8 February 6 2009 4 12 Section 27 Interval of Existence Interval of Existence Largest interval in which a solution of a first order ODE can be de ned Jiwen He University of Houston Math 3331 Section 19470 Lecture 8 February 639 2009 5 12 Section 27 Example Jiwen He University of Houston C2 O 130 SoV 1ac2dv1mtC gt 3 1t l C 30 2 10 330 i C 1330 Ixoli zzjot 10700 IoE OO71O Ifxo 0 gt 3615 O7 IOE 00700 0gtO olt0 ft7x x2 satisfies hypothe ses of EUT in any rectangle Unique solution for any 1130 x05 leaves any rectangle in finite time Solution is not defined for all re als if x0 7k 0 Math 3331 Section 19470 Lecture 8 February 6 2009 512 Section 27 A Existence When the RHS is Discontinuous 39Ex NP 1 2yft y0 3 ft 0 if tlt1 5 if t 2 1 t lt 1 2 3 2 2y gt yt 36 2t y For t gt 1 y1 36212 ii 4 Continue solution beyond 1 1 t 2 1 2y 5 211 362 2 z gt 38 2T 2t erSdt 1 52 3 5822 2T Combine 0 t f 3821 if t g 1 0 3 1 J 52 3 5822521 if t 2 1 o f is discontinuous at t 1 but unique solution exists for all t o y t is discontinuous at t 1 m Jiwen He University of Houston Math 3331 Section 19470 Lecture 8 February 6 2009 In Class Exercises Exercise 271 Ex 1 y 4y2 y0 1 Does IVP have a unique solution Yes because f 4 y2 and if83 2 2y are continuous everywhere Jiwen He University of Houston Math 3331 Section 19470 Lecture 8 February 6 2009 8 12 In Class Exercises Exercise 273 Ex 3 y t tan 1y y0 2 Does IVP have a unique solution Yes as Ex 1 Jiwen He University of Houston Math 3331 Section 19470 Lecture 8 February 6 2399 9 1239 In Class Exercises r Exercise 275 Ex 5 x tac l 1 330 2 0 Does IVP have a unique solution Yes because f and afaa tx 12 are continuous in any rectangle away from the horizontal line 00 1 and 9130 7E 1 Jiwen He University of Houston Math 3331 Section 19470 Lecture 8 February 6 2009 10 12 In Class Exercises 39 Exercise 277 EX 7 ty y t2 COSt y0 3 i Find general solution and sketch several solutions ii Show IVP has no solution and explain why this doesn t contradict EUT Answer i 3 yt tcost use integrating factor yttsintCt for ut exp 1tdt exp In t 1t 50 03271342 gt yt 2 cost gt yt sin t l C gt yt tsintCt gt 0 Answer ii Since y0 0 for any C there is no solution that satisfies y0 3 This doesn t con tradict EUT because f is not continuous at t O 30 20 10 0 10 20 30 50 Jiwen He University of Houston Math 3331 Section 19470 Lecture 8 February 6 2009 11 12 Read 288129 a Harwka 2 7 29 v M

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