M336 Differential Equations Lab 3 Numerical Solutions to Differential Equations By Euler's Method (Sections 1.4 in the Nagle/Saff/Snider text)

Applications of differential equations to problems, which have their basis in questions that arise from "real world" situations, frequently have no readily available closed form solutions. In these cases, approximations to the solutions are the best we can hope for. The simplest way to approximate a solution is to piece together the field lines of the differential equation. Our objectives in this section are as follows. 1. To learn how to numerically approximate solutions by means of Euler's Method and the Improved Euler's Method. 2. To see how the methods compare with each other and the actual solution if it is available. 3. To see how numerical schemes like Euler's method are valuable when actual analytic solutions are not readily available. Key Maple commands in this lab are the dsolve, proc and display commands.

Execute the following commands to restart and load plotting libraries: 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

Euler's Method Consider a differential equation of the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEjZHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y4USdub3JtYWxGJy1GIzYlLUYxNiVRJGR4dEYnRjRGN0Y6Rj0vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkkvJSliZXZlbGxlZEdRJmZhbHNlRidGOkY9 = f(x,y), y(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRkBGPg==) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRkBGPg==. To employ the Euler's Method, we begin by calculating the slope of the solution at the initial position, and then we use this slope to approximate the solution by a straight line. From here, we increase x by an incremental fixed amount, called the step size, and calculate the slope of the solution through the resulting point on the line, and then we repeat the process. To be precise, let us recall that the line segment joining two points 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 ) and 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 ) on a curve is called secant line. The slope of the secant line may be represented by 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. If (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) is a point on an unknown curve x(t), then the derivative LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEjZHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y4USdub3JtYWxGJy1GIzYlLUYxNiVRI2R4RidGNEY3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0Y6Rj0= at the point (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) can be used to approximate the coordinate (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). Setting the derivative equal to its approximate value, that is 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 provided 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 is close to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjUvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj1GQA==, and then clearing the fractions yields LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYvNiNRIUYnLUYjNictRi82JVEibkYnRjJGNS1JI21vR0YkNi1RIitGJy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkovJSlzdHJldGNoeUdGSi8lKnN5bW1ldHJpY0dGSi8lKGxhcmdlb3BHRkovJS5tb3ZhYmxlbGltaXRzR0ZKLyUnYWNjZW50R0ZKLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGWS1JI21uR0YkNiRRIjFGJ0ZGLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0ZGRjpGam5GRi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUZDNi1RKCZtaW51cztGJ0ZGRkhGS0ZNRk9GUUZTRlVGV0ZaLUYsNiVGLi1GIzYlRj9Gam5GRkZdb0ZqbkZG = 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 or equivalently, 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 where 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 . If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEjZHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y4USdub3JtYWxGJy1GIzYlLUYxNiVRI2R4RidGNEY3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0Y6Rj0= = f(x,y), then this formula becomes 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. This iteration formula together with the iteration 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 where h is the step size, is used to approximate points on the curve of the unknown function y(x) by Euler's method.

For example we can apply Euler's method to the initial value problem. 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 with initial condition y( 0 ) = 1. The above is separable and so can easily be solved by the method of separation of variables. The solution of this initial value problem is the exponential function with respect to variable x, 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. We can also ask Maple to verify this by using the command dsolve for solving differential equations. 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 However we will still illustrate the numerical method on this problem, and compare its accuracy to the actual solution. To numerically solve via Euler's method, we begin by defining the function representing the right hand side of the differential 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, and by declaring our known initial conditions for the starting values for the iteration, as well as the step size h for the iteration. 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GUy1JI21uR0YkNiRRJDAuMUYnRjk= Unless the solution is a straight line, the point (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) does not lie on the curve representing the true solution, but on the straight line tangent to the curve at the point (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). We approximate this as follows. 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 We citerate this process for the next five approximations: 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 The following commans form a table of the computed values, plot the points and the line segments connecting them, representing the approximate solution. 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 Here we display the true solution y = 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoZGlzcGxheUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRI3BwRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSNscEYnRi9GMkY9LUYsNiVRI3RwRidGL0YyRkEtRiw2I1EhRic= Why is the approximate solution always "underestimating" the true solution?

We can refine this procedure into a formal procedure as shown here. The following Maple procedure implements Eulers method -- place the cursor anywhere on the code and press Enter to this procedure defined to Maple. 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 We invoke the Eulers_Method procedure with an increment of .1 and halt after 20 steps as follows. Note that the method depends upon the function f already being defined. It also computes lists y and x as global variables -- meaning they are accessible once the procedure completes. These are lists store the approximated values 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 Define f before calling EulersMethod: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEieEYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGOS1GNjYtUSgmc3JhcnI7RidGOUY7Rj5GQEZCRkRGRkZIRmVuL0ZORmZuRmlu The four parameters (in order) in the call to EulersMethod below are initial values for x, y, h, and the number of iterations. 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 The following two commands create two plots Plot1 and Plot2 of the approximate and actual solution in blue and green, respectively -- To use display we create the two plot objects Plot1 and Plot2 first, and then give as arguments to display. 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 Now display simultaneously: 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 How good is the approximation? If we continue the approximation larger values of t, why does the error increase?

Exercise 1. Consider the initial value problem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEjZHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y4USdub3JtYWxGJy1GIzYlLUYxNiVRI2R4RidGNEY3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0Y6Rj0= = (3-y)y where y(0) = 2. Compare Euler's Method with the exact solution by taking h=.1 and n=20 interations. a) Graph the real solution and the approximate solution together on one graph and make a table listing the values of the actual solution and the approximate solution for the values 0, .1,.2,...,2. (Use the commands given below where you must supply the needed ???. ) b) Comment below on the (in)accuracy of the approximation: The first command below unassigns x and y from their values in the first example. 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 Have Maple solve the initial value problem DE: 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 Define the function representing the right hand side of the DE 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 Enter starting values for Euler's Method 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 Call Euler's Method: 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 Plot the solution returned by dsolve above for comparison to your approximate solution and then display together: 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 The display command will put both plots on the same axis. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEoZGlzcGxheUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUY2NiYtRiM2Jy1GLDYlUSZQbG90MUYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEmUGxvdDJGJ0YvRjIvJStleGVjdXRhYmxlR0ZJRkVGRS8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRidGaW5GRUZFLUZCNi1RIn5GJ0ZFRkcvRktGSUZMRk5GUEZSRlRGVi9GWkZYRmFvRmluRkU= Exercise 2. (a) Use the dsolve command to try to solve 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. What can you say about Maple's capability to solve differential equations in terms of elementary functions based on this result? LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Ki1GLDYlUSl1bmFzc2lnbkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUkjbW9HRiQ2LVEiJ0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRy8lKXN0cmV0Y2h5R0ZHLyUqc3ltbWV0cmljR0ZHLyUobGFyZ2VvcEdGRy8lLm1vdmFibGVsaW1pdHNHRkcvJSdhY2NlbnRHRkcvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR1EmMC4wZW1GJy1GLDYlUSJ4RidGNEY3Rj8vJStleGVjdXRhYmxlR0ZHRkNGQy1GQDYtUSI7RidGQ0ZFL0ZJRjZGSkZMRk5GUEZSL0ZVRlkvRlhRLDAuMjc3Nzc3OGVtRictRkA2LVEifkYnRkNGRUZIRkpGTEZORlBGUkZdb0ZXRjEtRjs2JC1GIzYnRj8tRiw2JVEieUYnRjRGN0Y/RmduRkNGQ0ZnbkZDRitGZ25GQw== 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 (b) Use Euler's method to numerically approximate the solution to (a) on the interval [1,2] with the initial value y(1) =1. Just plot the numeric approximation this time (since we don't have an explicit solution). LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=