Question

Suppose the autonomous DE in (2) has no critical points.Discuss the behavior of the solutions.

Short answer

The differential equation$f\left(y\right)=0$has no constant solutions. The solutions have neither upper nor lower bound and if$f\left(y\right)\ne 0$,the solutions$y\left(x\right)$ is either always decreasing or always increasing

Step-by-step solution

Step 1:Critical points

The zeros of the function f in $dy/dx=f\left(y\right)$are of special importance. Wesay that a real number c is a critical point of the autonomous differential equation $dy/dx=f\left(y\right)$if it is a zero of f—that is,$f\left(c\right)=0$ . A critical point is also called an equilibrium point or stationary point.

Step 2:Behavior of the solutions

If the differential equation has not critical points then there does not exists a value y such that.$f\left(y\right)=0$ This means the differential equation has no constant solutions. The solutions have neither upper nor lower bound.

Since $f\left(y\right)\ne 0$for all value of y then $f\left(y\right)$is either negative or always positive. This means the solutions $y\left(x\right)$is either always decreasing or always increasing.

So, it must be the case that the solution$y\left(x\right)$ ranges$(-\infty ,\infty )$.