Math 231: Differential Equations

1
Math 231 Differential Equations

• Set 3
• Working with Vector Fields
• Notes abridged from the Power Point Notes of
• Dr. Richard Rubin

2
Skinner's Constant

• That quantity which, when multiplied by, divided
  by, added to, or subtracted from the answer you
  get, gives the answer you should have gotten.

3
Direction Fields

• The text on p. 65 describes a situation that
  perhaps we all might want to share
• Drifting along a lake in boat, probably relaxing
  and letting the current carry us along!
• Of course, the text suggests you are reading a
  book about direction fields!

4
Direction Fields

• Definition
• A direction field is the set of tangent lines
  (slopes) of a given differential equation drawn
  in the xy-plane for selected points (xi,yi) of
  the differential equation.

5
Direction Fields

• Y axis points N X axis points E.
• In this coordinate system, our position is given
  parametrically by
• x(t) 10t and y(t) 10t
• Can you write y as a function only of x?

6
Direction Fields

• In this coordinate system, our position is given
  parametrically by
• x(t) 10t and y(t) 10t
• What function tells us the direction we travel at
  any instant in time?
• What is the derivative of y with respect to x?
• Chain Rule

7
Direction Fields

• In this coordinate system, our position is given
  parametrically by
• x(t) 10t and y(t) 10t
• If we draw a graph showing the direction of
  travel by a small arrow, we would get a picture
  like the one on pg 67.
• Since it shows direction of travel at each point
  in space, it is called a direction field.

8
Direction Fields

• x(t) 10t and y(t) 10t
• The graph on pg 67 is the direction field for our
  drifting boat.
• Suppose we start drifting from the origin of
  coordinate, what will be our approximate
  coordinates after 10 seconds? after 50 seconds?
• What equation describes our trajectory (y(x))?

9
Direction Fields

• x(t) 10t and y(t) 10t
• Suppose we change the scenario. Instead of
  drifting, we accelerate at 2 ft/sec2 while
  steering the boat due east starting from (0,0).
  The current is still carrying us north 10 ft and
  10 ft east each second.
• y(t) 10t
• as before. How do we determine x(t)?

10
Direction Fields

• We accelerate at 2 ft/sec2 while steering the
  boat due east starting from (0,0).
• The current is still carrying us north 10 ft and
  10 ft east each second.
• y(t) 10t
• as before. How do we determine x(t)?
• How do we find C?

11
Direction Fields

• We accelerate at 2 ft/sec2 while steering the
  boat due east starting from (0,0).
• The current is still carrying us north 10 ft and
  10 ft east each second.
• y(t) 10t
• as before. How do we determine x(t)?
• at t 0, dx/dt 10 before we accelerate!

12
Direction Fields

• We have
• We can find x(t) with the initial condition that
  x(0)0

13
Direction Fields

• The direction of travel is again given by dy/dx

14
Direction Fields

• The direction of travel is again given by dy/dx
• We can solve for t in terms of x

15
Direction Fields

• The direction of travel is again given by dy/dx
• Now

16
Direction Fields

• The direction of travel is again given by dy/dx
• Now

17
Direction Fields

• Suppose the solution passes thru a point (a,b) in
  the xy plane. Then, we have the condition that
• y(a) b.
• We know the slope at (a,b) is given by dy/dx. At
  (a,b), .

18
Direction Fields

• f(a,b) is the slope of the tangent to the
  solution at the point (a,b).
• We could choose (a,b) in such a way that we have
  a grid of equally spaced points in the xy plane.
• At each of these points, we plot slope of the
  solution by drawing an arrow with slope f(a,b).