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Sec. 21 First-Order Differential Equations 325
equation (2),
is of the form
where C is an arbitrary constant.
2. Direction field. The set of directions
tana = /(x, y)
is called a direction field of the differential equation (1) and is ordinarily
depicted by means of short lines or arrows inclined at an angle a.
Curves f(x, y) k, at the points of which the inclination of the field
has a constant value, equal to k, are called isoclines. By constructing the
isoclines and direction field, it is possible, in the simplest cases, to give a
Fig 105
rough sketch of the field of intagral curves, regarding the latter as curves
which at each point have the given direction of the field.
Example 1. Using the method of isoclines, construct the field of integral
curves of the equation
y'=*x.
Solution. By constructing the isoclines x~k (straight lines) and the direction
field, we obtain approximately the field of integral curves (Fig. 105).
The family of parabolas
is the general solution.
Using the method of isoclines, make approximate constructions of fields
of integral curves for the indicated differential equations:
2733. y' = x.
2734.