Calc 2 Lecture NotesSection 7.3Page 1 of 5

Section 7.3: Direction Fields and Euler's Method

Big idea:. If you can’t solve a differential equation explicitly, there are several ways to get a feel for the approximate solution: draw direction fields, obtain a numerical solution using Euler’s Method, and look at equilibrium solutions.

Big skill: You should be able to draw direction fields by hand and with Winplot, obtain a numerical solution using Euler’s Method by hand and with Excel, and compute equilibrium solutions and their stability by looking at the direction field.

Direction field: A graph indicating the slope of the tangent lines to the family of curves for a differential equation. This is useful, because you can “trace” the solution of the equation by following the arrows.

To draw a direction field in Winplot:


Practice:

  1. Solve the differential equation , then graph its direction field and several graphs from the family of curves. Notice how segments in the direction field close to the curves are tangent to the curve. Use the direction field to trace out a couple extra members of the family of curves.

x / y /
-2 / -2
-1 / -2
0 / -2
1 / -2
2 / -2
-2 / -1
-1 / -1
0 / -1
1 / -1
2 / -1
-2 / 0
-1 / 0
0 / 0
1 / 0
2 / 0
-2 / 1
-1 / 1
0 / 1
1 / 1
2 / 1
-2 / 2
-1 / 2
0 / 2
/
  1. Draw the direction field for , and draw approximate solution curves for the initial values y(-2) = -4, y(1) = -2, and y(-2) = 4.

x / y /
-2 / -2
-1 / -2
0 / -2
1 / -2
2 / -2
-2 / -1
-1 / -1
0 / -1
1 / -1
2 / -1
-2 / 0
-1 / 0
0 / 0
1 / 0
2 / 0
-2 / 1
-1 / 1
0 / 1
1 / 1
2 / 1
-2 / 2
-1 / 2
0 / 2
1 / 2
2 / 2
/

  1. Suppose that the differential equation is a model for the population of a species vs. time. Sketch the direction field and a few examples from the family of curves, then describe what those solutions represent for the population.

Euler’s Method:A technique for obtaining a numerical approximation to points on the curve of the solution of a differential equation.

We start by thinking about sub-dividing a region of x-values from a to b into n equal subdivisions, and integrating the differential equation using a left-endpoint numerical approximation:


Or, letting h = x, /

Practice:

  1. Compute an approximate solution for in the interval [1, 3], given the initial conditiony(1) = -2 and using a step size of x = 0.25.

i / xi / yi / f(xi, yi)
0 / 1 / -2
1
2
3
4
5
6
7
8

To perform Euler’s method quickly on your calculator:

  • Store the initial conditions in variables X and Y.
  • Enter the update formula for Y then for X separated by a colon.
  • Repeatedly Use 2ND ENTER until you are at the final x-value.
  • If you want to see the intermediate values of Y, you’ll have to type Y every time…
  • Example:

1  X

-2  Y

Y + (X + e^(-Y))*0.25  Y:X+0.25X

Y

2ND ENTER 2ND ENTER …

Equilibrium solutions: the function does not change (i.e., it is a constant). They are found by setting y = 0 and solving for y.

Astable equilibrium solution is a solution that other solutions tend to converge toward. Direction field lines point toward stable equilibriums.

An unstable equilibrium solution is a solution that other solutions tend to diverge away from. Direction field lines point away from unstable equilibriums.

Practice:

  1. Discuss the equilibrium solutions of .