SECTION 5.5 Differential Equations 305
solutiony =y(x). This results in the initial value problemof the
form
y′=p(x)y+q(x), y(a) =y 0.A good example of a commonly-encounterednonlinearODE is the
so-calledlogistic differential equation,
y′=ky(1−y), y(0) =y 0.5.5.1 Slope fields
A good way of getting a preliminary feel for differential equations is
through their slope fields. That is, given the differential equation
(not necessarily linear) in the form y′ = F(x,y) we notice first that
the slope at the point (x,y) of the solution curvey=y(x) is given by
F(x,y). Thus, we can represent this by drawing at (x,y) a short line
segment of slopeF(x,y). If we do this at enough points, then a visual
image appears, called the slope field of the ODE. Some examples
will clarify this; we shall be using the graphics softwareAutographto
generate slope fields.
Example 1. Consider the ODEy′=y−x. The slope field is indicated
below: