LESSON #

Stewart Section 10.2a - Direction Fields

The study of differential equations is about how to solve for a function or functions which satisfy a certain set of relationships between themselves and all of their derivatives.  For instance, the classic and simplified rate equation

is asking us to find the function y, whose derivative, or slope with respect to x,  is itself: y.  We know from learning about derivatives that the function whose derivative is itself is the exponential function....

y = ex + constant

and, in fact, it can be shown by using integration that this is indeed the solution to the above equation.  

Another example would be the equation

dy/dx = cos(x)

Again, in this case we are asking, what function y has, as its derivative, cos(x).  Knowing from our study of Calculus I that the derivative of sin(x) is cos(x), we can write that the function in question is

y = sin(x) + constant

Unless we are given some point on the line to pick what our constant is, our solution is not a single curve, but rather a family of curves.  

In some special cases like these, we can utilize what we know about integration to solve explicitly for the function y.  “Solving explicitly” or “solving analytically” means we can represent our answer in a nice neat function.  To prove what our intuition told us on equations (2) and (3), let’s solve these equations using a technique called “separation of variables”.  

In many more complicated differential equations, it is NOT possible to solve explicitly, so what should we do?  Consider a few examples from Section 10.2 of more complicated differential equations which cannot be solved explicitly.

dy/dx = x+y

dy/dx = x2 + y2 -1

L dI/dt + IR = E(t)

In this case, we must use what are called “numerical techniques”, methods which try to approximate the function which satisfies our equation, without actually trying to solve for the equation.  Section 10.2 of Stewart is about two important numerical approximation techniques, direction fields (aka slope fields) and Euler’s method.  Such numerical approximation techniques, many made infinitely easier by the advent of the computer, will enable us to understand the solution to our equation even if we cannot find an exact answer for our function.  

One such technique employs what are called “direction fields” or slope fields.   Because a differential equation tells us what the slope of our function must be for given values of x and y, we can use this information to sketch directional contours on our graph that will guide the behavior of our function.  The most efficient way of doing this correctly without using a computer is by constructing a T-table, where we solve for dy/dx, which is the slope of the function y, for different values of x and y.

In many cases, we will be able to take a limited number of points and use that to sketch our entire direction field.  In other cases, we will need to tabulate more points to get an accurate picture of our field.  Spreadsheet programs like Excel are excellent to do things like this.  I have included a

Figure from Stewart p. 608

IMPORTANT CONCEPTS

• Solving explicitly vs. solving numerically
• Family of curves as a solution to a differential equation
• Separation of Variables

PRACTICE PROBLEMS

1.  Match the direction field and the equation

2.  For each of the differential equations in the right hand column of Question 1, describe in words what the equation means.  Can you think of a physical situation which behaves like this?  

3.  The differential equation below describes a population of a species, P, in a particular area.  Their rate of birth (births per unit time) is given by b, and their rate of death (deaths per unit time).

(a) Draw the direction fields for this equation by using a calculation table for the cases where b-d > 0 and b-d < 0.  Describe the behavior of the population in each of these situations.  Are there parts of the direction field which should be excluded based on real-world considerations?  (hint: yes)

(b) An important consideration in solving differential equations is their “sensitivity to initial conditions”.  Depending on where we start our system, its behavior might be very different.  

ADDITIONAL INFORMATION

Slope Field Applet - http://www.math.ksu.edu/math240/java/lab1/lab1.html

This is a browser based applet that enables you to enter different differential equations, and will compute the slope field, or direction field, of a first-order differential equations.  This is what I used to generate the pictures above.