1. Calculus and Analytic Geometry II Cloud County Community College Spring, 2011 Instructor: Timothy L. Warkentin

2. Chapter 09: First-Order Differential Equations 09.01 Solutions, Slope Fields, and Euler’s Method 09.02 First-Order Linear Equations 09.03 Applications 09.04 Graphical Solutions of Autonomous Equations 09.05 Systems of Equations and Phase Planes

3. Chapter 9 Overview In Chapter 04 differential equations of the form were solved by integration to yield In Chapter 07 differential equations of the form were solved if they were separable - The exponential growth/decay differential equation is a famous example of a separable equation. In this chapter a solution is found for the general First-Order Linear Differential Equation

4. 09.01: Solutions, Slope Fields, and Euler’s Method 1 Verifying solutions of differential equations. Examples 1 & 2 If a differential equation has the form then f [x, y] is the slope of the solution y[x] + C. The general solution to a differential equation is a family of curves. A particular solution is determined when a point is specified through which one of the curves must pass. Slope Fields can be used to show the general shape of the solution family's curves. TI program SLOPEFLD – see handout folder. The relationship between a slope field and a particular solution. Example 2

5. 09.01: Solutions, Slope Fields, and Euler’s Method 2 Many times an exact solution to a differential equation is very difficult or even impossible to obtain. In these cases a numerical method that approximates the solution is used. One of the simplest and most common numerical approximations is given by Euler’s Method. Review of Linearization: finding the equation of the tangent line to y [x] at (x 0, y 0 ) when the slopes to y [x] are given by f [x 0, y 0 ].

6. 09.01: Solutions, Slope Fields, and Euler’s Method 3 The segment of L[x] between x 0 and x 0 +Δx is used to approximate y[x]. The error is Δy – ΔL. The process is then repeated.

7. 09.01: Solutions, Slope Fields, and Euler’s Method 4 To begin the process the following initial information is needed. Examples 3 & 4, TI program EULERT – see handout folder. –The initial point, (x 0, y 0 ) –The derivative function, f [x 0, y 0 ] –The step size, Δx –The interval over which the approximation is to be calculated or the number of approximation points desired.

8. 09.02: First Order Linear Equations 1 A first-order linear differential equation has the form. Example 1 Solving first-order linear differential equations by using an integrating factor. Examples 2 – 4 When does a first-order linear differential equation become a separable differential equation?

9. 09.03: Applications 1 Newton’s 1 st Law of Motion problems. F = m a, can be written as the differential equation: Example 1 Orthogonal Trajectory problems The original family of curves should be solved for the parameter so that it will become zero when the equation is implicitly differentiated. Example 2 Mixture problems. Example 3

10. 09.04: Graphical Solutions of Autonomous Equations 1 This section is not covered.

11. 09.05: Systems of Equations and Phase Planes 1 This section is not covered.