Page 157 - Advanced engineering mathematics

DIRECTION FIELDS EULER’S METHOD
TAYLOR AND MODIFIED EULER METHODS
CHAPTER 5

Approximation

of Solutions

In this chapter, we will concentrate on the first-order initial value problem

y = f (x, y); y(x 0 ) = y 0 .

Depending on f , it may be impossible to write the solution in a form from which we can
conveniently draw conclusions. For example, the problem
y − sin(x)y = 4; y(0) = 2

has the solution
x
y(x) = 4e −cos(x) e −cos(ξ) dξ + 2e 1−cos(x) .
0
It is unclear how this solution behaves or what its graph looks like.
In such cases, we may turn to computer-implemented methods to approximate solution val-
ues at specific points or to sketch an approximate graph. This chapter explores some techniques
for doing this.

5.1 Direction Fields

Suppose y = f (x, y), with f (x, y) given, at least for (x, y) in some specified region of the plane.
The slope of the solution passing through (x, y) is therefore a known number f (x, y).Forma
rectangular grid of points (x i , y j ). Through each grid point (x i , y j ), draw a short line segment
having slope f (x i , y j ). These line segments are called lineal elements. The lineal element through
(x i , y j ) is tangent to the solution through this point, and the collection of all the lineal elements
is called a direction field for the differential equation y = f (x, y). If enough lineal elements are

drawn, they trace out the shapes of integral curves of y = f (x, y), just as short tangent segments

drawn along a curve give an impression of the shape of the curve. The direction field therefore
provides a picture of how integral curves behave in the region over which the grid has been
placed.

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October 14, 2010 14:19 THM/NEIL Page-137 27410_05_ch05_p137-144