|Module 18 - Antiderivatives as Indefinite Integrals and Differential Equations|
|Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Lesson 4 | Self-Test|
|Lesson 18.4: Solving Differential Equations Graphically and Numerically|
Solutions to differential equations are found graphically and numerically in this lesson. Slope fields and particular solutions are also discussed.
The TI-89 has a graphing mode that allows you to find graphical and numerical solutions to differential equations. This mode is called the Differential Equation graphing mode.
Solve y' = 2x graphically.
The slope field contains short line segments with slope 2x at selected points in the window. This particular window uses 98 points arranged in 7 rows and 14 columns. The line segments could be thought of as linear approximations, or tangent lines, of particular solutions through the points.
Notice that all the line segments in a given column are parallel because they all have the same x-coordinate and therefore the same slope, 2x.
Finding a Particular Solution
Find the particular solution to the differential equation
The slope field is shown and the particular solution when y(2) = 1 is indicated by the darker connected curve.
The coordinates of t, x, and y will be displayed at the bottom of the screen.
18.4.1 Use the Trace feature to find y(3) for this particular solution.
Creating a Table of Values
You can use the TI-89 Table feature to create a table of values for the particular solution to y' = 2x and y(2) = 1. The table will display a numerical representation of the solution to the initial-value problem.
Each pair of values t and y1 represents a point (x, y) of the particular solution. Notice that the initial condition y(2) = 1 is shown and that y(3) = 6 is also there. You can display points on smaller intervals by adjusting tbl in the TblSet dialog box.
18.4.2 Graph the slope field and the solution for the initial-value problem y' = ex2 and y(-3) = 0.01. Use a [-3, 3] x [-0.5, 2] window. Remember to use t in place of x in the Y = Editor.
The Slope Field of y' = y
The slope field for the differential equation y' = y is significantly different from those previously shown. The solution of y' = y represents a function where the slope of the tangent at each point (x, y) is the y-coordinate of that point. The slope of solutions to y' = y is a function of y rather than x. How do you think this will affect the slope field?
18.4.3 Sketch a rough approximation of the slope field for y' = y. Compare your slope field with the slope field given by your TI-89. What is the major difference between the slope field for this problem and the previously illustrated slope fields?
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