M336 Laboratory Two First Order Differential Equations Graphical Analysis Using Direction Fields (Section 1.3 in the Nagle/Saff/Snider text) Restart starts a fresh session with Maple -- all previously stored values are erased.JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== Load the plots library to assist with plotting. LUkld2l0aEc2IjYjSSZwbG90c0dGJA== Load the differential equations tools library. LUkld2l0aEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSShERXRvb2xzR0Yn

We will look at first-order differential equations from a graphical point of view and we will make use of direction fields. Our objectives are as follows. 1. To learn how to use the computer to create the direction field for the differential equation. 2. To learn how the direction field gives pertinent information about solutions to differential equations. 3. To see how actual solutions fit onto the direction field. New command to be used in this section is the DEplot command.

A direction field for a differential equation is a two-dimensional plot comprised of tiny line segments that are portions of the tangent lines to solutions of the differential equation. Because explicit solutions to a differential equation cannot always be derived, direction fields are valuable in studying the behavior of solutions in these cases. A method known as the method of isoclines is a means of constructing the direction field for first order differential equations of the form 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. (See text page 20 --The construction is accomplished by setting the slope equal to various constants c and plotting the level curve 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 for those values for c. For each c, the slope of the solution y(x ) is the same.) Yet, even by using the method of isoclines, finding the direction field can be very labor intensive. Fortunately, in Maple a direction field can be constructed using either the DEplot command or the dfieldplot command.

Define the differential equation from your text figure 1.6a 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 and save it in the name ODE1_6a PkknT0RFMTZhRzYiLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YkNiNJInhHRiRGLSwmKiRGLSIiIyIiIkYqISIi Use the DEplot with appropriate options to plot the direction field for the differential equation: LUknREVwbG90RzYiNilJJ09ERTE2YUdGJC1JInlHRiQ2I0kieEdGJC9GKjshIiQiIiQvRihGLC9JJ2Fycm93c0dGJEklc2xpbUdGJC9JKGRpcmdyaWRHRiQ3JCIjN0Y2L0kmdGl0bGVHRiRJPERpcmVjdGlvbn5maWVsZH5vZn55Jz14XjIteUdGJA== Since the arrows at various points represent slopes of solutions curves that pass through those points, we can sketch various solutions cruves from the direction field. Why do we know that these curves will not intersect? (By Theorem 1 Section 2 ) By specifying an initial value for a solution curve, we will get a unique solution. We can ask Maple to plot along with the direction field the curves that satisfy certain initial conditions -- in the command below we ask for the four curves that satisfy y(0) = -2, y(0) = -1, y(0) = 1, y(0) = 2, respectively. 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

Below are similar commands to analyze the direction field for the differential equation of figure 1.7.a: 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. PkknT0RFMTdhRzYiLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YkNiNJInhHRiRGLSwkRiohIiM= LUknREVwbG90RzYiNilJJ09ERTE3YUdGJC1JInlHRiQ2I0kieEdGJC9GKjshIiMiIiMvRihGLC9JJ2Fycm93c0dGJEklc2xpbUdGJC9JKGRpcmdyaWRHRiQ3JCIjN0Y2L0kmdGl0bGVHRiQuLyoqSSpEaXJlY3Rpb25HRiQiIiJJJmZpZWxkR0YkRj1JI29mR0YkRj0tSSVkaWZmRyUqcHJvdGVjdGVkRzYkRihGKkY9LCRGKEYt LUknREVwbG90RzYiNipJJ09ERTE3YUdGJC1JInlHRiQ2I0kieEdGJC9GKjshIiMiIiMvRihGLC9JJ2Fycm93c0dGJEklc2xpbUdGJDwnNyQiIiFGNTckRjUiIiI3JEY1Ri03JEY1ISIiNyRGNUYuL0koZGlyZ3JpZEdGJDckIiM3Rj8vSSZ0aXRsZUdGJC4vKipJKkRpcmVjdGlvbkdGJEY3SSZmaWVsZEdGJEY3SSNvZkdGJEY3LUklZGlmZkclKnByb3RlY3RlZEc2JEYnRipGNywkRidGLQ==

Below we have the same for Figure 1.7.b: 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 PkknT0RFMTdiRzYiLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YkNiNJInhHRiRGLSwkKiZGKiIiIkYtISIiRjE= LUknREVwbG90RzYiNilJJ09ERTE3YkdGJC1JInlHRiQ2I0kieEdGJC9GKjshIiMiIiMvRihGLC9JJ2Fycm93c0dGJEklc2xpbUdGJC9JKGRpcmdyaWRHRiQ3JCIjN0Y2L0kmdGl0bGVHRiQuLyoqSSpEaXJlY3Rpb25HRiQiIiJJJmZpZWxkR0YkRj1JI29mR0YkRj0tSSVkaWZmRyUqcHJvdGVjdGVkRzYkRidGKkY9LCQqJkYnRj1GKiEiIkZG LUknREVwbG90RzYiNipJJ09ERTE3YkdGJC1JInlHRiQ2I0kieEdGJC9GKjshIiMiIiMvRihGLC9JJ2Fycm93c0dGJEklc2xpbUdGJDwkNyQkIiImISIiRi03JEYtJCIjP0YtL0koZGlyZ3JpZEdGJDckRjpGOi9JJnRpdGxlR0YkLi8qKkkqRGlyZWN0aW9uR0YkIiIiSSZmaWVsZEdGJEZESSNvZkdGJEZELUklZGlmZkclKnByb3RlY3RlZEc2JEYnRipGRCwkKiZGJ0ZERipGN0Y3 Why can we not use x = 0 for intial conditions for the above?

Next, consider the differential 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. At the Maple input prompt below enter the command to define the differential equation above and name it ODE1. Use t as the independent variable this time instead of x. JSFH Next enter the DEplot command to plot the direction field. Let the range of t and y be: t=-10..10,y=-11..10 JSFH Answer the following questions regarding what you see. (a) If a solution has the initial value of 4 at 0, that is y(0) = 4, what does it appear will be the behavior of the solution as t gets larger? As t gets smaller? Between what values will the solution y(t) always be for any choice of t? (You can click on the graph to see the coordinates of any point given in the top upper left hand corner.) (b) What do you notice about the direction of the direction arrows for any given value of y? From the differential equation (particularly the right hand side of it), is there good rationale for your answer? What is it? (c) If a solution had an initial value of y(0) = 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, what would be the general shape of the solution in terms of the solution with initial value of y(0) = 4? What is the reason for your answer when you consider the differential equation itself? (d) What are equilibrium solutions, that is, solutions curves that are constant -- i.e., of the form y = c? Check by actually plotting some of the solutions to see if your observations were correct. To do this we will enter some initial conditions into the DEplot command -- Use {[0,4],[0,4-2*Pi],[0,Pi/2] } JSFH Do you need to change any of your previous answers? -- If so do!

Before proceeding, execute the restart command. Consider the differential equation . 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 (a) Examine the direction field using the DEplot command and sketch various representative solutions in the window 0 <t<5 and -2<y<4. (b) If y(0)=1.9, what is the limit of y(t) at t goes to infinity? (c) If y(0)=2.1, what is the limit of y(t) at t goes to infinity? (d) Will any of the solution curves ever cross each other? Why? (e) If y is initially .5 (i.e. y(0) = .5), can y eventually be -1? Why?

Using DEplot, determine the direction field and some solution curves for each of the following differential equations in the window -5<t<5 and -5<y<5. Then answer the questions. JSFH (a) 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. Add various representative solutions curves. JSFH Are there any equilibrium solutions? Characterize the appearance of the solutions. (b) 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. Add various representative solution curves. JSFH Are there any equilibrium solutions? Characterize the appearance of the solutions. (c) 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 . Add various representative solution curves. JSFH Are there equilibrium solutions? Characterize the appearance of the solutions.