Equilibrium Solutions Examples

Given the following slope field, draw all equilibrium solutions:

Look for where the tangent lines are horizontal.

Determine all equilibrium solutions to the differential equation

.

In order for  to be zero, we must have y = 5. The constant function y = 5 is the only equilibrium solution.

Find all equilibrium solutions of the differential equation

Determine if each equilibrium solution is stable or unstable.

To find equilibrium solutions we set the differential equation equal to 0 and solve for y.

0 = y² – y = y(y – 1)

so the equilibrium solutions are y = 0 and y = 1.

Now to figure out if the other solutions are trying to snuggle up to or run away from each of these equilibrium solutions. When y > 1 the quantity

is positive, which means the slopes on the slope field will be positive when y > 1. Let's sketch this into our very rough slope field.

When y is between 0 and 1, y is positive but (y – 1) is negative, so the product y(y – 1) is negative. This means the slopes on the slope field will be negative when y is between 0 and 1.

Finally, when y is less than 0 both y and (y – 1) are negative, so the product y(y – 1) is positive. This means the slopes will be positive for y < 0.

The solutions above y = 1 are trying to get away from the equilibrium, so that's an unstable equilibrium. The other solutions are trying to get closer to y = 0, so that's a stable equilibrium: