Chap9_Sec2 - 9 DIFFERENTIAL EQUATIONS DIFFERENTIAL...

Info iconThis preview shows pages 1–17. Sign up to view the full content.

View Full Document Right Arrow Icon
DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS 9
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
DIFFERENTIAL EQUATIONS Unfortunately, it’s impossible to solve most differential equations in the sense of obtaining an explicit formula for the solution.
Background image of page 2
DIFFERENTIAL EQUATIONS Despite the absence of an explicit solution, we can still learn a lot about the solution through either: A graphical approach (direction fields) A numerical approach (Euler’s method)
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
9.2 Direction Fields and Euler’s Method In this section, we will learn about: How direction fields and Euler’s method help us solve certain differential equations. DIFFERENTIAL EQUATIONS
Background image of page 4
INTRODUCTION Suppose we are asked to sketch the graph of the solution of the initial-value problem y’ = x + y y (0) = 1 We don’t know a formula for the solution. So, how can we possibly sketch its graph?
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
INTRODUCTION Let’s think about what the differential equation means.
Background image of page 6
INTRODUCTION The equation y’ = x + y tells us that the slope at any point ( x, y ) on the graph (called the solution curve) is equal to the sum of the x - and y -coordinates of the point.
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
INTRODUCTION In particular, as the curve passes through the point (0, 1), its slope there must be 0 + 1 = 1.
Background image of page 8
INTRODUCTION So, a small portion of the solution curve near the point (0, 1) looks like a short line segment through (0, 1) with slope 1.
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
INTRODUCTION As a guide to sketching the rest of the curve, let’s draw short line segments at a number of points ( x, y ) with slope x + y .
Background image of page 10
DIRECTION FIELD The result is called a direction field. For instance, the line segment at the point (1, 2) has slope 1 + 2 = 3.
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
DIRECTION FIELDS The direction field allows us to visualize the general shape of the solution curves by indicating the direction in which the curves proceed at each point.
Background image of page 12
DIRECTION FIELDS Now, we can sketch the solution curve through the point (0, 1) by following the direction field as in this figure. Notice that we have drawn the curve so that it is parallel to nearby line segments.
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
DIRECTION FIELDS In general, suppose we have a first-order differential equation of the form y’ = F ( x, y ) where F ( x, y ) is some expression in x and y . The differential equation says that the slope of a solution curve at a point ( x, y ) on the curve is F ( x, y ).
Background image of page 14
SLOPE FIELD If we draw short line segments with slope F ( x, y ) at several points ( x, y ), the result is called a direction field (or slope field). These line segments indicate the direction in which a solution curve is heading. So, the direction field helps us visualize the general shape of these curves.
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
a.Sketch the direction field for the differential equation y’ = x 2 + y 2 – 1. b.Use part (a) to sketch the solution curve
Background image of page 16
Image of page 17
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 75

Chap9_Sec2 - 9 DIFFERENTIAL EQUATIONS DIFFERENTIAL...

This preview shows document pages 1 - 17. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online