If no arrows in a direction field point upwards, then the corresponding differential equation cannot have a stable equilibrium. True or false

Answer 1

False.

The direction field only gives information about the behavior of the solutions near a given point, but it does not provide information about the existence or stability of equilibria.

For example, consider the differential equation y' = -y^3. The direction field shows that all arrows point downwards, indicating that solutions decrease as we move to the right. However, this equation has a stable equilibrium at y=0.