MathHelp

Slope Fields and Solution Curves

In a Nut Shell: First order differential equations of the form

dy/dx = f(x,y)

don't always yield solutions for y(x) since the dependent variable, y, appears on both

sides of the equal sign. This d.e. may or may not be separable depending on the

expression, f(x,y).

However, the left side, dy/dx, represents the slope of the function, f(x, y). Thus a table

of values of x and y can be constructed to yield the slope of y(x) at each point (x,y).

Connecting nearby slopes with a "smooth" curve then produces a "solution curve" in the

"slope field" based on the starting point, (x_o,y_o). A family of curves yields a qualitative

interpretation of the solution for y(x) for various starting points, (x_o, y_o )

A separate class of differential equations, called autonomous, are of the form:

dy/dx = f(y)

Here the slope and the solution curves ( a smooth curve formed by connection of the slopes)

are independent of x. So a family of curves translated in x are also solution curves.

Strategy for construction of the Slope Field:

The slope field consists of plots of the slopes at individual points.

Step 1	Set up a table of x and y values with appropriate increments.
Step 2	Calculate f(x,y) for each x and y value.
Step 3	Plot the slopes.
Step 4	Connect slopes with smooth curves, the solution curves for the d.e.
Step 5	If the differential equation is autonomous, then the type of stability may be of interest. In this case plot a phase diagram to determine if the critical points are stable, unstable, or semi-stable. Then proceed to plot the slope field and solution curves.

Consider the following autonomous differential equation:

dy/dx = y² ( y² ˗ 4 )

Return to Differential Equations Notes