MA341 Maple Homework1 Due Thursday, January 17, 2008 at 11:34 pm Name: NORRIS LARRY Important: This assignment will be graded by a computer. In completing this homework you must follow the "Rules for Completing Your Maple Homework" that are given in the "Introductory Materials" section of the Calculus with Maple Homepage at the URL http://www.math.ncsu.edu/calculus . Below you will find a short summary of those Rules. Observe the following guidelines to maximize your score. Do not add or remove sections from the worksheet. Do not change the titles of the sections (Problem 1, Problem 2, etc.) Use only Maple 10.04 or Maple 11 Read all problems carefully and follow their instructions exactly. Make sure that your answers are assigned to the names specified in the problems. Make sure your commands work properly in sequence. You can check this by using the "Execute -> Worksheet" command under the Edit menu. Then look at your results to see if your commands generate any errors. If errors are present, fix them before submitting your worksheet, or comment them out. You must load any packages you need for each problem. For example, if you load plots in Problem 1, you must reload plots in Problem 2 if you need commands from the plots package.

(a) Consider the first order ODE 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 with the initial condition y(0) = 13 . Use Euler's method to estimate y(1) using 6 steps. Assign your answer to the name EulerEstimate. Use exact arithmetic throughout part (a). You can copy and paste the "Euler Script" from MA 341 Maple Basics Lesson 1. EulerEstimate:= put_your_answer_here ; (b) The error in Euler's method is the difference "exact value - approximate value". Use dsolve to find the exact solution of the given Initial value problem, and use the solution to find the error, as a 15 digit number, made in part (a) in using Euler's method to estimate the true value of y(1). Assign the error to TheEerror. TheError := put_your_answer_here ;

Use Euler's method with each of the following step sizes to estimate the value of y(.8e-1), where y is the solution of the initial-value problem diff(y(x),x) = 5*y(x)+8 , y(0) = 3 NOTE: If the symbol e-1 occurs anywhere in this problem together with a number and you do not know the meaning of e-1 (it is NOT exp(-1)) then you should read the topic "Maple's scientific notation " in the FAQ on the Calculus with Maple Homepage. You can copy and paste the "Euler Script" from MA 341 Maple Basics Lesson 1. (a) Step size h = .4e-1 . Assign your value of y(.8e-1) to ans1 and assign the list of points computed by Euler's method to listofpoints1. Be careful to include the initial data point in your listofpoints1. listofpoints1:= put_your_answer_here ; ans1:= put_your_answer_here ; (b) Step size h = .2e-1 . Assign your value of y(.8e-1) to ans2 and assign the list of points computed by Euler's method to listofpoints2. Be careful to include the initial data point in your listofpoints2. listofpoints2:= put_your_answer_here ; ans2:= put_your_answer_here ; (c) We know that the exact solution of the initial-value problem given above is y(x) = -8/5+23/5*exp(5*x). Use the Maple commands plot to plot the graph of y(x) = -8/5+23/5*exp(5*x) on the interval x = 0 .. .8e-1 . Then use Maple's pointplot command to plot the two lists of points. Then use Maple's display command plot two things on the same graph. Assign the plot with the plot of listofpoints1 and y(x) = -8/5+23/5*exp(5*x) to my_plot_1,and the plot with the plot of listofpoints2 and y(x) = -8/5+23/5*exp(5*x) to my_plot_2. with(plots): my_plot_1:= put_your_answer_here ; my_plot_2:= put_your_answer_here ; (d) Use your plot to decide whether your estimates in parts (a), (b) and (c) are underestimates or overestimates. Assign your answer to ans4 below by replacing the words put_your_answer_here in the string with ONE of the words underestimate or overestimate. ans4:= put_your_answer_here ; (e) The error in Euler's method is the difference "exact value - approximate value". Find the errors, as 16 digit numbers, made in part (a) in using Euler's method to estimate the true value of y(.8e-1), namely 5.262393609149842 . Assign the error for h= .4e-1 to error1 and assign the error for h= .2e-1 to error2 . error1:= put_your_answer_here ; error2:= put_your_answer_here ;

(a) Consider the first order ODE 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 with the initial condition y(0) = 17 . Use Euler's method to estimate y(1) using 9 steps. Assign your answer to the name EulerEstimate. Use exact arithmetic throughout part (a). You can copy and paste the "Euler Script" from MA 341 Maple Basics Lesson 1. EulerEstimate:= put_your_answer_here ; (b) The error in Euler's method is the difference "exact value - approximate value". Use dsolve to find the exact solution of the given Initial value problem, and use the solution to find the error, as 11 digit numbers, made in part (a) in using Euler's method to estimate the true value of y(1). Assign the error to TheEerror. TheError := put_your_answer_here ;

(a) Consider the first order ODE 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 with the initial condition y(0) = -15 . Use Euler's method to estimate y(1) using 7 steps. Assign your answer to the name EulerEstimate. Use exact arithmetic throughout part (a). You can copy and paste the "Euler Script" from MA 341 Maple Basics Lesson 1. EulerEstimate:= put_your_answer_here ; (b) The error in Euler's method is the difference "exact value - approximate value". Use dsolve to find the exact solution of the given Initial value problem, and use the solution to find the error, as 17 digit numbers, made in part (a) in using Euler's method to estimate the true value of y(1). Assign the error to TheEerror. TheError := put_your_answer_here ;