Page 174 - Schaum's Outline of Differential Equations

CHAPTER 18

Graphical and

Numerical Methods for

Solving First-Order

Differential Equations

QUALITATIVE METHODS

In Chapter 2, we touched upon the concept of qualitative methods regarding differential equations; that is,
techniques which are used when analytical solutions are difficult or virtually impossible to obtain. In this
chapter, and in the two succeeding chapters, we introduce several qualitative approaches in dealing with
differential equations.

DIRECTION FIELDS

Graphical methods produce plots of solutions to first-order differential equations of the form

where the derivative appears only on the left side of the equation.

1
Example 18.1. (a) For the problem / = -y + x + 2, we have/(jt, y) = -y + x + 2. (b) For the problem / = y + 1, we have
2
f(x, y) = y + 1. (c) For the problem y' = 3, we have (x, y) = 3. Observe that in a particular problem, (x, y) may be
f
f
independent of x, of y, or of x and y.
Equation (18.1) defines the slope of the solution curve y(x) at any point (x, y) in the plane. A line element
is a short line segment that begins at the point (x, y) and has a slope specified by (18.1); it represents an approxi-
mation to the solution curve through that point. A collection of line elements is a direction field. The graphs of
solutions to (18.1) are generated from direction fields by drawing curves that pass through the points at which
line elements are drawn and also are tangent to those line elements.
If the left side of Eq. (18.1) is set equal to a constant, the graph of the resulting equation is called an
isocline. Different constants define different isoclines, and each isocline has the property that all line elements

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