Question

What is an equilibrium solution to a differential equation?

Answer 1

Hint: In this type of problem, we have to understand the concept of equilibrium solution to a DE whose derivative should be zero. Equilibrium solution looks like a horizontal line. And we can also understand the concept of slope field because you can find the equilibrium solution by finding a horizontal line that fits into the slope field.

Complete step by step answer:
First of all, we need to understand the concept of differential equation that means
The differential equation is an equation containing one or more derivatives, where derivatives are terms describing the rate of change of quantities that vary continuously. In general, the solution of the differential equation is an equation that expresses the functional dependence of one variable or more than one variable.

Now, we understand the concept of equilibrium solution. An equilibrium solution is a solution to a DE whose derivative is zero everywhere. On a graph an equilibrium solution looks like a horizontal line. Given a slope field, you can find equilibrium solutions by finding everywhere a horizontal line fits into the slope field. Equilibrium solutions come in two flavours: stable and unstable.These terms are easiest to understand by looking at slope fields.

A stable equilibrium solution is one that other solutions are trying to get to. If we pick a point a little bit off the equilibrium in either direction, the solution that goes through that point tries to snuggle up to the equilibrium solution. An unstable equilibrium solution is one that the other solutions are trying to get away from. If we pick a point a little bit off the equilibrium, the solution that goes through that point is trying to run away from the equilibrium solution.

Note: In this type of problems, solutions on one side of an equilibrium solution move towards the equilibrium solution and on the other side of the equilibrium solution move away from it we call the equilibrium solution semi-stable, and $$y=-1$$ is a semi-stable equilibrium solution.