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## Differential Equations

by: Otilia Murray I

56

0

6

# Differential Equations MAT 022B

Otilia Murray I
UCD
GPA 3.88

Staff

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COURSE
PROF.
Staff
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Class Notes
PAGES
6
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KARMA
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## Popular in Mathematics (M)

This 6 page Class Notes was uploaded by Otilia Murray I on Tuesday September 8, 2015. The Class Notes belongs to MAT 022B at University of California - Davis taught by Staff in Fall. Since its upload, it has received 56 views. For similar materials see /class/187452/mat-022b-university-of-california-davis in Mathematics (M) at University of California - Davis.

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Date Created: 09/08/15
Midterm 1 Review7 MAT 22B Hillel Raz January 247 2009 The following is a study guide for MAT 22B7 midterm 1 It is a list of topics covered that could be on the exam The list of problems is longer than would be asked in a 50 minute exam7 but it provides you with more sample problems so you can practice Note that all examples from class7 the book and homework problems should also be reviewed The exam will be closed book and notes and no calculator Ch 1 Direction elds know how to draw a direction eld and how to interpret your graph If there is a direct dependence on 257 what happens to the graph7 or to y in the limit as t a 00 Examples 1y 4y6 2y yy3 3y ye t 4y y27t23 Know the different types of differential equations ordinary DE7 partial DE7 linear vs non linear and know how to tell the order of a DE Examples Classify the following DE s according to the above classi cations and determine the order of each 51 62y 4 y St 61 cost sint2y y 71m uyy Umy um Be able to verify that a given function is a solution of a differential equation by plugging in Similarly7 solve for a parameter that makes a given function a solution of a DE Examples Verify that the following functions are solutions of the given DE7 825 my 4y 0 25gt 0 y1t t 3420 t zlnt 91 y 0 y1t 00805 24205 00805 i 811105 Can you suggest another possible solution for number 9 For y 5quot what values of r make the following DE true 103 7 3y 23 0 Ch 2 Different methods of solving DE s that is7 integrating factor method and separation of variables Know how to recognize when to use either For integrating factors I Make sure equation is in correct form my glttgt ll Find f ftdt lll Multiply equation both sides by 5f tldt lV Rewrite left hand side of the equation using reverse power rule should become gafftdt39 V lntegrate and solve for y or whatever other variable is being used Note you can always check if your solution is correct by plugging it into the DE and seeing if its true ie is the left hand side still equal to the right hand side Examples Draw a direction eld for the following then solve the DE s Draw your solution on the direc tion eld it should 7 ow7 through the arrows ie go in the same direction 114 y 5sin2t y0 1 12y y1 4 an an 131 55 y0 1 14ty 7 y 295475 gt 0 y0 2 Models There are lots of models Know the different types we covered in class and in the homework ow interest population models etc Go with Steven Pon s method DUDES Draw a picture Use units Declare variables Express derivative in terms of other stuff Solve For population models and other models graphs are very bene cial Make sure you understand how to draw and interpret the phase diagram equilibrium points and stability Of course the equation needs to be autonomous meaning there is only one variable written L y not L 100572011 L N For ow problems set up the DE remembering that the change ow in ow out Usually you are asked to check concentration of something hence you ll most likely need to divide the rate out part by the volume of the container as that affects the concentration especially if that volume is constantly changing A picture if of utmost importance here Examples look at any in the book in section 23 1 6 are ow problems for instance Don t skip section 25 thoughl Try to set them up that s the hardest part Existence and uniqueness know what the theorems say 241 242 281 and be able to explain in your own words what proving existence and uniqueness means Examples For the following determine the interval in which a solution is certain to exist 154 7 t2y 2ty 3t2 3473 1 16lnty 1 6y sint y2 3 Euler s method Know the method how to set up and how to evaluate a few iterations writing out the solutions without evaluating the actual numbers is ok you won t be able to use a calculator on the exam Examples Use Euler s method with h 01 to nd approximate solutions for the given initial value problem for t 17 13 Do you think the solutions are converging or diverging A direction eld might be helpful in answering the second question 17 y 74xy3 y1 2 181 ity 01337 y0 1 Know the method of successive approximations Be able to set it up and determine the formula for the nth iteration Examples Determine gtnt for arbitrary n 19 y 2y2 y0 0 20447523717 y0 0 Know the difference between a DE and a difference equation Know how to solve a dif ference equation and nd its equilibrium solutions Examples Solve the given difference equations in terms of the initial value yo 21 yn1 08yn 22yn1 71n1yn Get enough sleep and eat a good meal the night before the exam MAT22B71 Review Problems for Final Exam Table 1 1 t y Laplace Transforms 1 1 g s gt 0 2 5 E s gt a 3 t 7 n gt 0 integer 93 s gt 0 4 sinat 3 gt 0 5 cos at m s gt 0 6 sinhat 3 gt lal 7 coshat 5212 3 gt M 8 eatsinbt s gt a 9 eatcosbt sgta 10 We 71 gt 0 integer 7 3 gt a 11 Ma snFltsgt e s 1f07 f 10 12 07705 FONS Table 2 Simple Trigonometric Integrals 1 fseczmdztanz0 2 fcsczzdzicotz0 3 fsecmtanz dzseczC 4 fcsczcotzdzicscm0 5 fseczdzlnlsecztanml0 6 fcsczdzilnlcsczcotml0 7 f arcsinmC 8 f 22 arctanz C Table 3 Some Trigonometric Identities H sinA i B sinAcosB i sinBcosA cosA i B cosAcosB sinAsinB OJ sinA i sinB 2sin g cos g q cosA cosB 200s cos cosA 7 cosB 72sin sin The Final Exam includes all topics covered in the course The break down of the content is a follows between 40 to 45 is from Chapter 7 between 35 to 40 is from Chapters 3 and 6 between 20 to 25 is from Chapters 1 and 2 Study tips 1 Review the Midterms and the Practice Midterms 2 Review the problems given below 3 Review the homeworks and 4 Review the lectures For the following problems7 nd the general solution and draw the phase portrait 1 00 to 72 i4 i7 ltgt lt gtltgt For review on nonhomogenous systems7 the problems from 79 on your HW set are good Review the mixing problem 22 in 71 A hard boiled egg is removed from the boiling water and placed in an ice bath to cool Assume that the temperature ofthe icebath is always at 00C and the initial temperatures ofthe egg yolk and the egg whites are 550C and 450C respectively Suppose Oz 2 is the conductivity constant of the membrane between egg yolk and B 3 is the conductivity constant of the shell Let T1 and T2 denote the temperature of the yolk and whites7 respectively Apply Newton s Law of Cooling to set up the system of ODE s then nd the temperatures T1 and T2 as functions of time t in seconds 10 Consider the following two vector functions x1t lt i x2t lt In what intervals does the linear combination Xt 01X1t 62X2t represent the general solution of a system of rst order linear ODE s 7

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