Slope Fields
Objective: To find graphs and
equations of functions by the use
of slope fields.
Slope Fields
• If we interpret dy/dx as the slope of a tangent line,
then at a point (x, y) on an integral curve of the
equation dy/dx = f(x), the slope of the tangent line is
f(x). What is interesting about this is that the slopes
of the tangent lines to the integral curve can be
obtained without actually solving the differential
equation.
Slope Fields
dy
 x2 1
dx
• For example, if
then we know without
solving the equation that at the point where x = 1 the
tangent line to an integral curve has slope 12  1  2
and more generally, at a point where x = a the
tangent line to an integral curve has slope a 2  1 .
Slope Fields
• A geometric description of the integral curves of a
differential equation dy/dx = f(x) can be obtained by
choosing a rectangular grid of points in the xy-plane,
calculating the slopes of the tangent lines to the
integral curves at the gridpoints, and drawing small
portions of the tangent lines through those points.
The resulting picture, which is called a slope field or
direction field for the equation, shows the “direction”
of the integral curves at the gridpoints. With
sufficiently many gridpoints it is often possible to
visualize the integral curves themselves.
Slope Fields
• For example, we see a slope field for the differential equation
dy/dx = x2. We also can look at the same field with the
integral curves imposed on it-the more gridpoints that are
used, the more completely the slope field reveals the shape of
the integral curves.
Slope Fields
• Look at the following equations and slope fields. Match the
differential equation with the slope field and sketch the
solution curves through the highlighted points.
a) dy  2 b) dy   x c) dy  x 2  4 d) dy  e x / 3
dx
dx
dx
dx
Slope Fields
• Look at the following equations and slope fields. Match the
differential equation with the slope field and sketch the
solution curves through the highlighted points.
a) dy  2 b) dy   x c) dy  x 2  4 d) dy  e x / 3
dx
b
dx
d
dx
dx
c
a
Slope Fields
• Look at the following equations and slope fields. Match the
differential equation with the slope field and sketch the
solution curves through the highlighted points.
b) dy   x
dx
Slope Fields
• Look at the following equations and slope fields. Match the
differential equation with the slope field and sketch the
solution curves through the highlighted points.
dy
d)
 ex/3
dx
Slope Fields
• Look at the following equations and slope fields. Match the
differential equation with the slope field and sketch the
solution curves through the highlighted points.
c) dy  x 2  4
dx
Slope Fields
• Look at the following equations and slope fields. Match the
differential equation with the slope field and sketch the
solution curves through the highlighted points.
a)
Functions in Two Variables
• We will continue to look at first-order differential
equations. We first looked at equations of the form
y/ = f(x). For example, y /  x3 expresses the
derivative in the variable x alone. However, the
equation y /  sin( xy) is of the form y/ = f(x, y),
expressing the derivative in both x and y.
Example 1
• Sketch the slope field for the differential equation
f ( x, y )  y  x
Example 1
• Sketch the slope field for the differential equation
f ( x, y )  y  x
Example 2
• Sketch the slope field for y = xy/4 for the gridpoints
where -1 < x < 1 and -1 < y < 1.
Euler’s Method
• Now we will develop a method for approximating the
solution of an initial-value problem of the form
y /  f ( x, y ); y ( x0 )  y0
• We will not attempt to approximate y(x) for all values
of x; rather we will choose some small increment x
and focus on approximating the values of y(x) at a
succession of x-values spaced x units apart, starting
from x0 . We will denote these x-values by
x1  x0  x
x2  x1  x
x3  x2  x
Euler’s Method
• The technique that we will describe for obtaining
these approximations is called Euler’s (pronounced
Oiler’s) Method. Although there are better
approximation methods available, many of them use
Euler’s Method as a starting point, so the underlying
concepts are important to understand.
Euler’s Method
• The basic idea behind Euler’s Method is to start at the
known initial point (x0, y0) and draw a line segment in
the direction determined by the slope field until we
reach the point (x1, y1) with x-coordinate x1  x0  x .
If x is small, then it is reasonable to expect that this
line segment will not deviate
much from the integral curve
y = y(x). We will repeat the
process using the slope field
as a guide.
Euler’s Method
• The initial value and x will always been given, so the
x-values are very easy to find. We need to find the
corresponding y-values. We will use the idea that the
slope of a line is defined as m  y , and rewrite the
x
equation as y  mx .
Euler’s Method
• The initial value and x will always been given, so the
x-values are very easy to find. We need to find the
corresponding y-values. We will use the idea that the
slope of a line is defined as m  y , and rewrite the
x
equation as y  mx .
• The slope that we will need will be the value of dy/dx
evaluated at the previous point. We multiply this
slope by x and this will give us y . We add this to
the y-value before and we have our new y-coordinate.
Example 3
• Use Euler’s Method with a step size (x ) of 0.1 to
make a table of approximate values of the solution of
the initial-value problem y /  y  x; y(0)  2 .
Example 3
• Use Euler’s Method with a step size (x ) of 0.1 to
make a table of approximate values of the solution of
the initial-value problem y /  y  x; y(0)  2 .
• The initial point is (0, 2).
• x = 0.1
• We need to find y .
Example 3
• Use Euler’s Method with a step size (x ) of 0.1 to
make a table of approximate values of the solution of
the initial-value problem y /  y  x; y(0)  2 .
• The initial point is (0, 2).
• x = 0.1
• We need to find y .
• We will evaluate y/ at the point (0, 2) and multiply by x
y  m x
y  (2  0)(0.1)  .2
Example 3
• Use Euler’s Method with a step size (x ) of 0.1 to
make a table of approximate values of the solution of
the initial-value problem y /  y  x; y(0)  2 .
• The initial point is (0, 2).
• x = 0.1, y = .2
• The next point is (.1, 2.2)
• We will repeat this process for each additional point,
now evaluating y/ at the point (.1, 2.2)
y  m x
y  (2.2  0.1)(0.1)  .21
Example 3
• Use Euler’s Method with a step size (x ) of 0.1 to
make a table of approximate values of the solution of
the initial-value problem y /  y  x; y(0)  2 .
• The initial point is (0, 2).
• x = 0.1, y = .21
• The next point is (.1, 2.2), (.2, 2.41)
• We will repeat this process for each additional point,
now evaluating y/ at the point (.2, 2.41)
y  m x
y  (2.41  0.2)(0.1)  .221
Example 3
• Use Euler’s Method with a step size (x ) of 0.1 to
make a table of approximate values of the solution of
the initial-value problem y /  y  x; y(0)  2 .
• The initial point is (0, 2).
• x = 0.1, y = .221
• The next point is (.1, 2.2), (.2, 2.41), (.3, 2.631)
• We will repeat this process for each additional point,
now evaluating y/ at the point (.3, 2.631)
y  m x
y  (2.631  0.3)(0.1)  .2331
Example 3
• Use Euler’s Method with a step size (x ) of 0.1 to
make a table of approximate values of the solution of
the initial-value problem y /  y  x; y(0)  2 .
• The initial point is (0, 2).
• x = 0.1, y = .2331
• The next point is (.1, 2.2), (.2, 2.41), (.3, 2.631),
(.4, 2.8641)
• We continue for as long as the problem requires.
Homework
• Pages 584-585
• 1, 3, 6, 7
• Section 8.3