would like to book a lecturer for an econ maths quiz. topics:Differential Equations &Difference EquationsNUMBER OF QUESTIONS WILL BE 4
Microsoft Word - ECON2016 Unit 6 and 7 Practice Questions.docx Unit6AdditionalPracticeQuestions Find additional practice questions for Unit 6 in preparation for the graded quiz. Solutions will not be posted but you can email me if you have any specific queries. 1. Findthesolutiontotheinitialvalueproblems. (i) (ii) (iii) 2. Usetheintegratingfactormethodtosolvethedifferentialequations. (i) (ii) 3. Solvethesecondorderdifferentialequations. (i) (ii) (iii) 4. Coalisextractedfromamineataratethatisproportionaltotheamountof coalthatremainsinthemine.ShowthatifRistheamountthatremainsafter tyearsand istheinitialamountofcoalintheminethen ,wherekisaconstant. Giventhatafter10years,80%oftheinitialamountofcoalremains, (i) findthevalueoftheconstantk,and (ii) findthetimetakenfor50%oftheinitialamountofcoaltoremain. 2)1(,2 -== y y x dx dy 1)0(, == - yxe dx dy y 1)1(, == yxye dx dy x 3)1(,3 3 ==- yx x y dx dy 0)1(;)1( ==++ - yeyx dx dyx x xey dx dy dx yd 3 2 2 34 -=++ 0 when 0 and 1 ,65 22 2 ===-=++ - x dx dyyey dx dy dx yd x 0 when 3 and 1 ,2232 2 ====++ - t dt dQQeQ dt dQ dt Qd t 0R kteRR -= 0 5. ThepopulationP(t),attimet,ofaprolificbreedofrabbitsissuchthatthe rateofgrowthofitspopulationisproportionaltothesquareofits population,sothat (i) Showthat , where isthepopulationattime . (ii) “Doomsdayisthetimewhen .Arguewhydoomsday occurswhen . (iii) Giventhat andthatthereare4rabbitsafter4months, estimatethevalueoftheconstantkintheaboveequation.When doesdoomsdayoccur? 6. Attimetthepopulationoftheworldisxwherexistreatedasacontinuous variable.Thetime-rateofincreaseofxduetobirthsis andthetime-rate ofdecreaseofxduetodeathsis . (i) Arguewhythedifferentialequationrelatingxandtis ,where areconstants. (ii) Giventhat andtisinyears,showthatthe populationoftheworldwillbedoubledinapproximately35years. 7. Attime ,Osamaopenedanaccountanddeposited$8,000inabank.At timetdays,theinterestrateis2%ontheamountPthatispresent.Osama withdraws,onadailybasis,$100. (i) TreatingPasacontinuousvariable,showthat (ii) SolvethedifferentialequationtofindPintermsoft. (iii) Findthetimetakenfortheaccounttohaveabalanceof$11,000. (iv) Whentheaccountbalancewasshowing$11,000,Osamadecidesto withdraw$xonadailybasis.Writedown,intermsofx,anew differentialequationthatissatisfiedbyx. (v) Showthatif ,thenthebalanceamountwilldecreaseona dailybasis. .0 where2 >= kkP dt dP tkP P tP 0 0 1 )( - = 0P 0=t ¥®)(tP 0 1 kP t = 20 =P xa xb x dt dx )( ba -= ba and 04.0 and 06.0 == ba 0=t .500050 -= P dt dP 220>x 8. Solvethefollowingsystemofdifferentialequations. subjectto . 7043 ,3592 21 2 21 1 +--= +-= yy dt dy yy dt dy 1)0(,0)0( 21 == yy ECON2016_u6-20190312-v1 ECON2016: Mathematical Methods of Economics II • UNIT 6 _ 20190225_v1 1 UNIT 6 Differential Equations Overview Any situation where the quantities change over time can be modelled using differential equations. As a result, differential equations have become indispensable tools in the modelling and solution of problems in modern society. This unit explores finding solutions to first order and second order differential equations as well as systems of differential equations. Learning Objectives By the end of this unit, you will be able to: 1. Solve first order differential equations. 2. Solve second order differential equations. 3. Solve systems of differential equations. ECON2016: Mathematical Methods of Economics II • UNIT 6 _ 20190225_v1 2 This unit is divided into two sessions as follows: Session 6.1: First Order Differential Equations Session 6.2: Second Order Differential Equations Session 6.3: Systems of Differential Equations Readings & Resources Note to Students: Sometimes hyperlinks to resources may not open when clicked. If any link fails to open, please copy and paste the link in your browser to view/download the resource. Required Readings Hoy, M., Livernois, J., McKenna, C., Rees, R., & Stengos, T. (2011). Mathematics for economics. MIT Press. Differential Equations https://www.youtube.com/watch?v=zid7J4EhZN8 Khan Academy: Differential Equations https://www.khanacademy.org/math/ap-calculus-ab/ab-diff- equations/ab-diff-eq-intro/v/differential-equation-introduction Khan Academy: Method of Undetermined Coefficients https://www.khanacademy.org/math/differential-equations/second-order- differential-equations/undetermined-coefficients/v/undetermined- coefficients-1 Khan Academy: Second Order Differential Equations https://www.khanacademy.org/math/differential-equations/second-order- differential-equations Second Order Linear Homogeneous Differential Equations https://www.youtube.com/watch?v=UyCwAFQt4v0 Systems of Differential Equations http://tutorial.math.lamar.edu/Classes/DE/HOSystems.aspx ECON2016: Mathematical Methods of Economics II • UNIT 6 _ 20190225_v1 3 Session 6.1 First Order Differential Equations Introduction This session discusses how to solve first order differential equations. It focuses on different types of first order differential equations such as: ordinary, linear, non-linear and separable. First Order Differential Equations: Part 1 Let us start out our discussion on differential equations by reviewing some of what you may already know. For example, let us find the equation of the curve whose gradient is given by !" !# = 2? and passes through (1,3). We have !" !# = 2? (Differential equation (DE)) ? =∫2??? ? = ?+ + ? (General solution) The graph representing this equation is shown below. Note that C is a constant and can take on any value. Figure 6.1 ECON2016: Mathematical Methods of Economics II • UNIT 6 _ 20190225_v1 4 Suppose we have specific values for x and ? . Then we can find values for C. For example, if ? = 1and? = 3 gives then 3 = 1+ + ? 3 = 1 + ? ? = 2 So the equation is ? = ?+ + 2 Most times we want to express how a variable changes over time, t , where time is considered to be a continuous variable. e.g. !"!4 + ? = 7 Definitions Let us formally define some key expressions that we will use in our discussion. Ordinary Differential Equation contains ordinary derivatives as opposed to partial derivatives. We can also have partial differential equations e.g. !" !4 = 3?+? Differential Equations of Order i. !" !# = 2? 1st Order ii. ! 7" !47 + 2 !" !4 + ? = 2 2nd Order iii. ! 8" !48 + 2 !" !4 = 4 3rd Order Linear or Non-Linear Differential Equations !" !4 + ?:? = ? Linear !" !4 + ?+ = ? Non-Linear Separable Differential Equations The example shown above is an example of a separable differential equation. ECON2016: Mathematical Methods of Economics II • UNIT 6 _ 20190225_v1 5 Example 1 Solve !"!# = 2? Separate variables ?? = 2??? Integrate both sides ∫?? = ∫ 2??? ? = ?+ + ? Example 2 Solve !"!4 = 4;:47 "7 , ? = 6when? = 0 Separate variables ??? = (? + 3?+)?? Integrate both sides ∫ ?+?? =∫(? + 3?+)?? " 8 : = 4 7 + + ?: + ? multiply through by 3 ?: = :4 8 + + 3?: + 3? ?: = :4 8 + + 3?: + ? (General Solution) Now, ? = 0???? = 6, this gives 6: = ? 216 = ? So ?: = :4 7 + + 3?: + 216 (Particular Solution) ? = F:4 7 + + 3?: + 216 8 ECON2016: Mathematical Methods of Economics II • UNIT 6 _ 20190225_v1 6 Example 3 Solve !"!4 = 2? Separate variables !" " = 2?? Integrate both sides ∫ !" " = ∫2?? ln|?| = 2? + ? ? = ?+4;J ? = ?+4. ?J ? = ??+4 where ? = ?M constant Example 4 Solve !"!4 = 4" 47;N Separate variables !"4 = 4 47;N ?? Integrate both sides ∫ !" " = ∫ 447;N ?? ln|?| = N+ ln(? + + 1) + ln|?| ln|?| = ln √?+ + 1 + ln|?| ln|?| = lnP?√?+ + 1P ? = P?√?+ + 1P ECON2016: Mathematical Methods of Economics II • UNIT 6 _ 20190225_v1 7 First Order Differential Equations: Part 2 Let us now explore more differential equations and see how we can deal with those that are not separable. Integrating Factors Example 1 Solve !"!4 − 3? = ?? :4; ?(0) = 4 Here we cannot separate variables so we need an integrating factor (I.F.) We can write a general first order linear differential equation as !" !4 + ?(?)? = ?(?) In our example, ?(?) = −3 and ?(?) = ??:4 To solve this, we multiply by the integrating factor (I.F.) ?∫U(4)!4. Note that we ignore the constant of integration, C. The I.F. turns the left-hand side into the derivative of a product. Now let us solve the problem. !" !4 − 3? = ??:4,?(0) = 4 …….(*) First, find I.F. Here ?(?) = −3 So I.F. = ?∫W:!4 = ?W:4 Next multiply (*) by I.F. to get ?W:4 !" !4 − 3?W:4? = ??:4. ?W:4 ?W:4 !" !4 − 3?W:4? = ? ! !4 (?W:4?) = ? ECON2016: Mathematical Methods of Economics II • UNIT 6 _ 20190225_v1 8 Finally, integrate both sides, ∫ !!4 (?