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# Class Note for MATH 3331 with Professor He at UH

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This 18 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 11 views.

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Date Created: 02/06/15

cog 5 in 31mm mWw Wmquot mm 039 me vzmbb wmevm mm mm m New Fwy my ODES men zuse mm m 09 m my as new mg m 5 we gem my Ex mm 2nd m m w n 73 w swam n y m We m min mammogth m It y 5 am my be m mm m M m4 m w the We on my cog HM min mvmz m A Wmquot We mm cog v WW 5 2 dmmm mmquot yt m w m we mu 9 m we mm whee m 5 gm 5m m ytt1 or s 2 mmquot a M m4 myquot 0 Section 21 General Solution and Solution Curves Show that yt Cequot2 is a solution of y 2ty 4 General Solution The solution formula yt Cet2 which depends on the arbitrary constant 4 4 C describes a family of solutions and is called a general solution Solution Curves The graphs of these solutions drawn in the figure are called solution curves I Jiwen He University of Houston Math 3331 Section 19470 Lecture 2 January 23 2009 6 18 Shwih21yt1rt gemmw y y a w 2 vimzuhvmmimn gamma cm WWW Wm in W m Mme we mummme mm w mame Ewan A may cog mmgm in mm mm y my ytnm 5 a 2n miu mm mm knmy w v39 A Wmquot We w s 2 Mama ngan m m m a m ILyt mu t Wm mmmgmwm m 5 mm m rayon w mumquot m z W 7 We m mnnmew y ya mm a Section 21 Interval of Existence Interval of Existence The interval of existence of a solution to an IVP is defined to be the largest interval over which the solution can be defined and remain a solution Find the interval of existence for the solution to the IVP 2 01quot y y with y0 1 U1 AK mu Jiwen He University of Houston Math 3331 Section 19470 Lecture 2 January 289 2009 9 18 Section 21 Example y o Show that yt 2 Ce t is a solution of y 2 y for any constant C lt4 it 0 Find the solution that satisfies the initial condition y0 1 o What is the interval of v existence of this solution 4 Ill Jiwen He University of Houston Math Section 19470 Lecture 2 January 23 2009 10 18 at ya be m uimn owe cog yIt y m N We Wm yt mug 2 mm M runny mm m on we mm m you m m wemmm mmquot w w you mummy the ws 5 ihes v mmesumm ma 27d We nus mus m whilhesbve m Mn Section 21 39v Direction Field Direction Field for y fty Draw a line segment with slope ftyj attached to every grid point tyj in a rectangle R where fty is defined Rtyla t bandc y d The result is called a direction field 8 6 439 O 2 generated the direction a 0 v v 39 r 4 y y 639 8 I I I I I I I I I 2 15 1 05 0 05 1 15 2 Jiwen He University of Houston Math 3331 Section 19470 Lecture 2 January 23 2009 Section 21 It 39 u 39339 Definition of ODE Solutions IVP Geometric Inter Geometric interpretation of Solutions Direction field provides information about qualitative form of solution curves Finding a solution to the differential equation is equivalent to the geometric problem of finding a curve in ty plane that is tangent to the direction field at every point 2 MATLAB generated the z o 0 g g g g h h solution curve of y y y01 g 439 E 39 l J l U l l O U Q O J l 15 Jiwen He University of Houston Math 3331 Section 19470 Lecture 2 January 23 2009 13 18 D Section 21 r quot39i 39 oflVP Numerical Solution Euler s Method of the Solution of IVP y fty yt0 yo 1 Plot the point P0t0y0 2 Move a prescribed distance along a line with slope ft0y0 to the point P1t1y1 3 Continue in this manner to produce an approximate solution curve of the IVP 4 U 6 5 4 MATLABgeneratedan 39 39 P 2 y y7 Y0 1 1P Math 3331 Section 19470 Lecture 2 January 23 Jiwen He University of Houston 2009 ln Class Exercises u Exercise 24 Exercise 24 1 Show that the given solution is a general solution of the differential equation y y2t yt2t 2 l Cet C 3 23 2 Use a computer or calculator to sketch members of the family of solutions for the given values of the arbitrary constant 3 Experiment with different intervals for t until you have a plot that shows what you consider to be the most important behavior of the family I Jiwen He University of Houston Maih 3331 sectionilrg a Ledle 2 i January 232009 i 157 1g ln Class Exercises gm 7 Exercise 213 Use the given general solution to find a solution of the differential equation having the given initial condition Sketch the solution the initial condition and discuss the solutions interval of existence ty yt2 yt13t2Ct y12 1O yt 13t2 53t The interval of existence is 0 Cgt0 5 12 1 2 E Ill Jiwen He University of Houston Math 3331 r Section 19470 Lecture 2 January 23 2009 15 18 3 ln Class Exercises i Exercise 219 Exercise 219 Plot the direction field for the differential equation by hand 2 l y t tany2 Do this by drawing short lines of 1 the appropriate slope centered at each of the integer valued coordinates ty where 2 t 2and 1 y 1 Jiwen He University of Houston Math 3331 Section 19470 Lecture 2 January 23 20097 17 18 Read 22 a Harwka 2 2 742472 Lecture 2 521an 2 Dnevenha mmquot and swam 1W He Wm M WM WW M mm m KVLuNnvmmzn

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