Old Exam I Take Home Problem 1 Examples

PROBLEM 1:  EULER'S METHOD PROBLEM (Take-Home Test Problem):  Solve the differential equation given below analytically, finding the particular solution satisfying the given condition.  Approximate the solution using Euler's Method over the interval [0,pi] with a step size of pi/32.  Discuss the nature of the accuracy of Euler's Method as applied to this problem.  If the Euler solutions are too high or too low at some sub-interval and then seem to become more accurate explain why this happened.  Draw a graph showing both the analytical solution and the discrete points depicting the Euler's Method solution over the interval [0,pi].  In the 2067 problem x_o= 0, y_o = 2, and h = pi/32.  Show the computations involved in finding y₁ and y₂.  Click here to see a development of Euler's Method and some examples.  Here is an online utility from Vanderbilt University that will do the calculations for you in using Euler's Method and produce a table of values (this one was down the last time I tried it).  Here is a very nice Euler's Method applet by David Protas of California State University that will both draw a graph of the Euler solution and generate a table of values.  DO THE 2067 PROBLEM. 

Exact and Euler Solution Graphs using Excel for the Term 2041 take-home problem (dy/dx = 6x² - 24x + 22     y(0) = 1) can be found by following the link.

2047 PROBLEM:          dy/dx = 8x² - 26x + 20     y(0) = 1

2054 PROBLEM:          dy/dx = -3x²  + 12x - 8     y(0) = 4 

2057 PROBLEM:          dy/dx = 3cos(x) + 1    y(0) = 1

2064 PROBLEM:          dy/dx = y[cos(x)]    y(0) = 1

2064 Bonus:  Do the same problem over the same interval with a step size of pi/16.

More 2064 Bonus:  Do the same problem with a step size of pi/8 but over the interval [0,83pi/4].  What is happening to the y-value in Euler's solution as x gets very large?  Even More Bonus:  Why is this happening?  Hint  Hint2 Hint3

2067 PROBLEM:  dy/dx = 4y[sin(4x)]    y(0) = 2

PROBLEM 1:  EULER'S METHOD PROBLEM (Take-Home Test Problem):  Solve the differential equation given below analytically, finding the particular solution satisfying the given condition.  Approximate the solution using Euler's Method over the interval [0,4pi] with a step size of pi/8.  Discuss the nature of the accuracy of Euler's Method as applied to this problem.  If the Euler solutions are too high or too low at some sub-interval and then seem to become more accurate explain why this happened.  Draw a graph showing both the analytical solution and the discrete points depicting the Euler's Method solution over the interval [0,4pi].  In the 2074 problem x_o= 0, y_o = 2, and h = pi/8.  Show the computations involved in finding y₁ and y₂.  Click here to see a development of Euler's Method and some examples.  Here is a very nice Euler's Method applet by David Protas of California State University that will both draw a graph of the Euler solution and generate a table of values.

2074 PROBLEM:          dy/dx = 4cos(2x) + 1     y(0) = 2

PROBLEM 1:  EULER'S METHOD PROBLEM (Take-Home Test Problem):  Solve the differential equation given below analytically, finding the particular solution satisfying the given condition.  Approximate the solution using Euler's Method over the interval [0,9pi/4] with a step size of pi/8.  Discuss the nature of the accuracy of Euler's Method as applied to this problem.  If the Euler solutions are too high or too low at some sub-interval and then seem to become more accurate explain why this happened.  Draw a graph showing both the analytical solution and the discrete points depicting the Euler's Method solution over the interval [0,9pi/4].  In the 2077 problem x_o= 0, y_o = 0, and h = pi/8.  Show the computations involved in finding y₁ and y₂.  Click here to see a development of Euler's Method and some examples.  Here is a very nice Euler's Method applet by David Protas of California State University that will both draw a graph of the Euler solution and generate a table of values.

2077 PROBLEM:          dy/dx = sin(x) - y    y(0) = 0

PROBLEM 1:  EULER'S METHOD PROBLEM (Take-Home Test Problem):  Solve the differential equation given below analytically, finding the particular solution satisfying the given condition.  Approximate the solution using Euler's Method over the interval [0,9pi/4] with a step size of pi/8.  Discuss the nature of the accuracy of Euler's Method as applied to this problem.  If the Euler solutions are too high or too low at some sub-interval and then seem to become more accurate explain why this happened.  Draw a graph showing both the analytical solution and the discrete points depicting the Euler's Method solution over the interval [0,4pi].  In the 2084 problem x_o= 0, y_o = 2, and h = pi/8.  Show the computations involved in finding y₁ and y₂.  Click here to see a development of Euler's Method and some examples.  Here is a very nice Euler's Method applet by David Protas of California State University that will both draw a graph of the Euler solution and generate a table of values.

2084 PROBLEM:          dy/dx = x/3 + 4cos(x)    y(0) = 2

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        Lane Vosbury, Math Chair, Seminole Community College   email:  vosburyl@scc-fl.edu

        This page was last updated on 10/08/08          Copyright 2002          webstats