Chap9_Sec2

# Chap9_Sec2 - 9 DIFFERENTIAL EQUATIONS DIFFERENTIAL...

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DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS 9

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DIFFERENTIAL EQUATIONS Unfortunately, it’s impossible to solve most differential equations in the sense of obtaining an explicit formula for the solution.
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)

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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
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?

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INTRODUCTION Let’s think about what the differential equation means.
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.

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INTRODUCTION In particular, as the curve passes through the point (0, 1), its slope there must be 0 + 1 = 1.
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.

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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 .
DIRECTION FIELD The result is called a direction field. For instance, the line segment at the point (1, 2) has slope 1 + 2 = 3.

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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.
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.

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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 ).
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.

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a.Sketch the direction field for the differential equation y’ = x 2 + y 2 – 1. b.Use part (a) to sketch the solution curve
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Chap9_Sec2 - 9 DIFFERENTIAL EQUATIONS DIFFERENTIAL...

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