MATH 2250 Linear Algebra and Differential Equations
Chapter 1: First-Order Differential Equations
1.3 Slope Fields and Solutions Curves
Given a first-order differential equation
),( yxf
dx
dy
,
the function f(x,y) is a formula for the slope of a solution curve y at any point (x,y) in the domain
of f.
Depending on the nature of the function f(x,y), we may be able to solve for y explicitly as a
function of x, say y(x). In many cases, we may obtain an implicit relationship between y and x in
which we can only view y as a function of x on a restricted region of the xy plane.
This section addresses the use of the differential equation itself to sketch the slope field of a
solution curve.
The diagram at right is the
slope field for
yx
dx
dy
.
Notice that the slopes are zero
along the line y = x and, in fact,
are constant along ALL lines
with slope m = 1.
We may use the slope field to
sketch graph of a solution curve
passing through a given point
(x,y) by following a short line
segment at the point to its other
end and following a new line
segment whose slope corresponds
to the new point. In this manner,
we may construct a piecewise-
linear approximation solution.
Sketch solutions to
yx
dx
dy
given that (a) y(-2) = 1 and (b) y(−2) = −3.
Notice that any two distinct solutions will not intersect and that all solutions become asymptotic
to the solution y = x. (WHY?)
