Calculus and Analytic Geometry II

Cloud County Community College

Spring, 2011

Instructor: Timothy L. Warkentin

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

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

][xfdx

dy

ykdx

dy

],[ yxfdx

dy

.][][ dxxfxy

].[][ yhxgdx

dy

].[][ xQyxPdx

dy

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

],[ yxfdx

dy

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 (x0, y0) when the slopes to y [x] are given by f [x0, y0].

000000 ],[],[][ xyxfyxyxfxL

09.01: Solutions, Slope Fields, and Euler’s Method 3

• The segment of L[x] between x0 and x0+Δx is used to approximate y[x]. The error is Δy – ΔL. The process is then repeated.

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, (x0, y0)– The derivative function, f [x0, y0]– The step size, Δx– The interval over which the approximation is to be calculated or

the number of approximation points desired.

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?

][][ xQyxPdx

dy

09.03: Applications 1

• Newton’s 1st 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

.dt

dvmF

09.04: Graphical Solutions of Autonomous Equations 1

• This section is not covered.

09.05: Systems of Equations and Phase Planes 1

• This section is not covered.