MATH202-19S2 Assignment A1Due — 4pm, 23 AugustYour completed assignment should be handed in via the MATH202 box in the reception area onLevel 4 of the Erskine building. You may do the assignment on your own, or as a pair with oneother student. If you do the assignment as a pair then you will both get the same mark. Ensurethat your name(s) and student ID number(s) are clearly written on the assignment.The breakdown of marks is: A1.1: 20%, A1.2: 30%, A1.3: 30%, presentation: 20%.To get the presentation marks your assignment must be clearly readable, written in completesentences, all figures should be clearly labelled and referenced in your text, and the actual questionsmust be answered! You should conclude with statements like: “The general solution tothe differential equation given in part (b) is therefore .....”. It is not necessary to type yourassignment. You can use Maple or any other symbolic maths software to evaluate integrals(just explain that this is what you have done).Note: there are two parts in the first question, four parts in the second question and three partsin the third question.A1.1 Solve the following problems using the techniques required in each question(i) Given thatshow that, by reduction of order, the general solution to this equation isy(t) = c1 et + c2 (1 + t).(ii) Find the general solution oft y00 (1 + t) y0 + y = tusing variation of parameters.A1.2 (i) Consider a chemical reaction in which compounds A and B combine to form athird compound X. The reaction can be written asIf 2g of A and 1g of B are required to produce 3g of compound X, then theamount of compound x at time t satisfies the differential equation1where a and b are the amounts of A and B at time 0 (respectively), and initiallynone of compound X is present (so x(0) = 0). Time is in units of minutes, andk is the reaction rate, per minute per gram.Use separation of variables (and integration by partial fractions) to show thatthe solution can be expressed in the formwhere the constant c depends on k, a, b. Now suppose that a = 15g, b = 20gand after 10min, 15g of compound X has been formed. Using (?) or otherwise,find the amount of X after 20mins. Show you working, and write down yourreasoning in simple sentences.(ii) With the same values a = 15, b = 20, use the geometric method to identifythe ranges of x for which x = x(t) is an increasing function and a decreasingfunction. Sketch the solution, and determine how much of compound X willform in the limit as t → ∞. Your answer should be illustrated with suitablediagrams. Clear hand-drawn diagrams are perfectly acceptable.(iii) Now use the values k = 0.009731, a = 15 and b = 20 and ode45 in MATLAB toproduce a numerical solution to equation (*) over the interval t ∈ [0, 30], withthe initial condition x(0) = 0. Your answer should include a printout of a graphof this solution, suitably labelled, and an explanation of how you calculated it.(iv) Repeat your geometric and numerical methods with different initial concentrationsof A and B. Look in particular at (a, b) = (15, 40) and (a, b) = (40, 15).How much of compound X is formed as t → ∞ in each of these two cases?Finally, give a general formula for how limt→∞ x(t) depends on a and b and givea brief physical interpretation.In presenting your answers to this part it is not necessary to give as much detail ofthe calculations as in the earlier questions. But do continue to write in completesentences!A1.3 We will revisit the pendulum problem that you approximated in Lab 2,N + sin(θN ) = 0, 0 θN (0) = a, θ0N (0) = 0.The subscript N means that we are considering the nonlinear equation. Remember thatθN measures the position of the pendulum (in radians) with respect to the vertical.(i) We will start with an approximation to problem (1). It is known that if θN issmall we can approximate the nonlinear problem with its linear version, Where the L subscript means that we consider the linear problem. Find thesolution for (2) in terms of a.2(ii) Use the code your wrote in Lab 2 to approximate the solution to (1) using Heun’smethod (or modified Euler, or RK-2) for a = 0.1, 0.2, 0.3, · · · , 1. You shoulduse h = 0.001. Plot in the same figure θL(t) and θN (t) with t ∈ [0, 2π] forvalues of a 0.1, 0.4, 0.7 and 1. Explain what you see.(iii) Calculate the relative error between θN and θL with respect to a and plot yourresults. The x-axis should have the ten values of a and the y-axis should havethe relative error between the two variables at each value of a. Remember thatthe relative error is defined byeR =||θN θL||||θN || .Use the function norm in MATLAB to calculate the norms. When do you thinkthe linear approximation to the full nonlinear problem starts to lose validity?3本团队核心人员组成主要包括BAT一线工程师，精通德英语！我们主要业务范围是代做编程大作业、课程设计等等。我们的方向领域：window编程 数值算法 AI人工智能 金融统计 计量分析 大数据 网络编程 WEB编程 通讯编程 游戏编程多媒体linux 外挂编程 程序API图像处理 嵌入式/单片机 数据库编程 控制台 进程与线程 网络安全 汇编语言 硬件编程 软件设计 工程标准规等。其中代写编程、代写程序、代写留学生程序作业语言或工具包括但不限于以下范围:C/C++/C#代写Java代写IT代写Python代写辅导编程作业Matlab代写Haskell代写Processing代写Linux环境搭建Rust代写Data Structure Assginment 数据结构代写MIPS代写Machine Learning 作业 代写Oracle/SQL/PostgreSQL/Pig 数据库代写/代做/辅导Web开发、网站开发、网站作业ASP.NET网站开发Finance Insurace Statistics统计、回归、迭代Prolog代写Computer Computational method代做因为专业，所以值得信赖。如有需要，请加QQ：99515681 或邮箱：99515681@qq.com 微信：codehelp QQ：99515681 或邮箱：99515681@qq.com 微信：codehelp
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