7Idirectionflds


 * Direction Fields**

Direction Fields (or Slope Fields) are a graphical way of understanding 1st Order D.E.s

They are very useful for D.E.s that can't be solved using calculus. Many D.E.s that have more than one variable in the function can't be solved using Year 12 level calculus. A lot can't be solved with any form of calculus.

These are functions in the form: math \dfrac{dy}{dx}=f(x,\,y) math

Constructing Direction Fields

To construct a direction field, consider the coordinates (x, y) of a point. The D.E. gives the gradient of a solution at that point. Draw a short line segment at those coordinates with that gradient. Repeat for a large number of points and a direction field will emerge.

A curve can then be drawn starting from some initial coordinates so that the line segments in that region are parellel to {tangent to} the curve being drawn. The resulting curve is a solution to the D.E.

Repeating this for a number of different initial coordinates produces a family of curves which are all solutions to the D.E. Notice that the family of curves will often not all have the same shape.

Example

For the differential equation: math \dfrac{dy}{dx}=x+y math the gradient at a point (x, y) is given by adding the ordinates x + y

Draw a short line segment with gradient x + y at the point (x, y) Repeat for a large number of points and a direction field emerges (see right)

A particular solution can then be graphed by using an initial condition such as (0, 1) and then plotting the curve which runs tangent to the line segments in that region. (see below left)

Particular solutions with different initial conditions have quite a different shape in this example (see below right)

Direction Fields on the Classpad

On the **Main Menu**, Select the **DiffEqGraph** Option.

In the DiffEq tab, enter the differential equation (eg: y' = x + y) Tap the graph icon and a direction field will be drawn.

You may need to adjust the domain and range (3rd icon from left) (I find Steps = 10 to give an appropriate appearance)

The IC tab allows you to enter Initial Conditions OR The 4th icon from the left (circled) allows you to tap the graph at a point and get a particular solution which passes through that point.

Analysing Direction Fields

A typical question may give you a direction field. You draw in a solution and suggest the appropriate equation for a solution to the D.E. which produced that direction field.

Example In this case, there appears to be a vertical asymptote at x = 0.

The curve rises from negative infinity at x = 0 and then continues with a positive but reduced gradient as x increases.

From your knowledge of graphs (or by sketching each of the suggested answers on your calculator) it should be clear that E is the correct answer { y = log e (x) }