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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 QyYtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMmSShTdHVkZW50R0YnNiNJMk51bWVyaWNhbEFuYWx5c2lzR0YoISIiLUkqaW50ZXJmYWNlR0YlNiMvSStydGFibGVzaXplR0YoIiNLIiIi

Euler's Method Consider a differential equation of the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEjZHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y4USdub3JtYWxGJy1GIzYlLUYxNiVRI2R4RidGNEY3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0Y6Rj0= = f(x,y), y(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R0Y9RkBGPg==) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R0Y9RkBGPg==. 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 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 = 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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. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnZHNvbHZlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiQtRjY2Ji1GIzYrLUYsNiVRInlGJ0YvRjItSSNtb0dGJDYtUSInRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHUSYwLjBlbUYnLUZCNi1RIj1GJ0ZFRkdGSkZMRk5GUEZSRlQvRldRLDAuMjc3Nzc3OGVtRicvRlpGam5GPi1GQjYtUSIsRidGRUZHL0ZLRjFGTEZORlBGUkZUL0ZXRmVuL0ZaUSwwLjMzMzMzMzNlbUYnRj4tRjY2JC1GIzYjLUkjbW5HRiQ2JFEiMEYnRkVGRUZmbi1GaG82JFEiMUYnRkVGRS8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRictRiw2I1EhRidGRQ== 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 equationLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KC1JJm1mcmFjR0YkNigtRiM2JS1GLDYlUSNkeUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRj1RJ25vcm1hbEYnLUYjNiUtRiw2JVEjZHhGJ0Y5RjxGP0ZCLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZOLyUpYmV2ZWxsZWRHUSZmYWxzZUYnLUkjbW9HRiQ2LVEiPUYnRkIvJSZmZW5jZUdGUy8lKnNlcGFyYXRvckdGUy8lKXN0cmV0Y2h5R0ZTLyUqc3ltbWV0cmljR0ZTLyUobGFyZ2VvcEdGUy8lLm1vdmFibGVsaW1pdHNHRlMvJSdhY2NlbnRHRlMvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Ziby1GIzYnLUYsNiVRImZGJ0Y5RjwtRlU2LVEwJkFwcGx5RnVuY3Rpb247RidGQkZYRlpGZm5GaG5Gam5GXG9GXm8vRmFvUSYwLjBlbUYnL0Zkb0ZecC1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRInhGJ0Y5RjwtRlU2LVEiLEYnRkJGWC9GZW5GO0ZmbkZobkZqbkZcb0Zeb0ZdcC9GZG9RLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRjlGPEY/RkJGQkY/RkJGK0Y/RkJGK0Y/RkI=, 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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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYjLUkjbW5HRiQ2JFEiM0YnL0YzUSdub3JtYWxGJ0Y+LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGTC8lKXN0cmV0Y2h5R0ZMLyUqc3ltbWV0cmljR0ZMLyUobGFyZ2VvcEdGTC8lLm1vdmFibGVsaW1pdHNHRkwvJSdhY2NlbnRHRkwvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zlbi1GRzYtUSJ+RidGPkZKRk1GT0ZRRlNGVUZXL0ZaUSYwLjBlbUYnL0ZnbkZcb0YrLUY2NiYtRiM2Iy1GOzYkUSIyRidGPkY+RkBGQy1GRzYtUSIrRidGPkZKRk1GT0ZRRlNGVUZXL0ZaUSwwLjIyMjIyMjJlbUYnL0ZnbkZpby1GLDYlUSJoRidGL0Yy 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 The following commands 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?

Maple has a procedure called Euler that implements Eulers method --it is part of the Student[NumericalAnalysis] package that must be loaded first.

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

Exercise 1. Consider the initial value problem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEjZHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y4USdub3JtYWxGJy1GIzYlLUYxNiVRI2R4RidGNEY3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0Y6Rj0= = 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 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 so that we can use them in DE below: 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 Have Maple solve the initial value problem DE -- use dsolve JSFH Now invoke Euler's method: JSFH JSFH Exercise 2. (a) Use the dsolve command to try to solve 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. First unassign x and y: 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 Now use dsolve: JSFH (b) Use Euler's method to numerically approximate the solution to (a) on the interval [1,2] with the initial value y(1) =1. Use a step size of 0.1 LUkmRXVsZXJHNiI2Jy8tSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkieUdGJDYjSSJ4R0YkRi4tSSRleHBHNiRGKUkoX3N5c2xpYkdGJDYjLCQqJEYuIiIjISIiLy1GLDYjIiIiRjsvRi5GNi9JKW51bXN0ZXBzR0YkIiM1L0knb3V0cHV0R0YkSSxpbmZvcm1hdGlvbkdGJA== Exercise : Complete Exercise 10, page 28; Find by trial and error the value of h for Euler's method such that 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 is approximated to within +/- 0.01 of y(x), for the initial value problem: 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. Also find to within +/- 0.05 the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1JJW1zdWJHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtSSNtbkdGJDYkUSIwRicvRjtRJ25vcm1hbEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0ZDLyUvc3Vic2NyaXB0c2hpZnRHRkJGRUZDRitGRUZD such that 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. Compare your answers with those given by the actual solution 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. Have Maple solve the initial value problem DE: 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 Invoke Euler's Method with increasingly smaller step sizes (half the size each time -- so double the number of steps) until the approximate solution y at x = 1 has not changed by more than +/-0.01 Start with h = 0.5 Step size h = 0.5 LUkmRXVsZXJHNiI2Jy8tSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkieUdGJDYjSSJ4R0YkRi4sJkYuIiIiRishIiIvLUYsNiMiIiFGNS9GLkYwL0kpbnVtc3RlcHNHRiQiIiUvSSdvdXRwdXRHRiRJLGluZm9ybWF0aW9uR0Yk JSFH Once you have found a satisfactory value for h and corresponding number of steps, plot the Euler approximation along with true solution JSFH