Question

I had asked this question last week:

State whether or not the following equations are autonomous and identify all equilibrium solutions (if any): y' = t-1. Clearly it is not autonomous. The book says that it has no equilibrium solutions, isoclines are t=constant. However, in Chegg, the same question has been answered saying that t=1 is an equilibrium solution. Please clarify this discrepancy.

Someone answered by showing some steps and then concluded, therefore, t=1 is an equilibrium solution for equation y' = t-1

I am not convinced with this response and I had put my comment: The answer in the 'solution manual' says that there is no equilibrium solution. A solution to a differential equation is the function y(t). How can t=1 be a solution when we expect the solution to be some y value? Also if you graph the slope field, it is horizontal line only for t=1, not for all t. This doesn't explain long term behavior. Please clarify.

I didn't get any response to my comment above. Can someone please explain why the solution is t=1 considering my argument above?

Answer 1

Consider the given differential equation,

Equilibrium solutions are the solutions of the type .

And the isoclines are the curves given by where is a constant.

Since, the solution of the given differential equation is and but none of them is an equilibrium solution.

(The other solution   is the solution of the equation after integration of ).

As equilibrium solution is of the type and none of the solution are of this type, there are no equilibrium solution.

But, the isocline exists which is .