Asymptotic Behavior of the Solution

The given differential equation dy/dx = ay - by^2, where a and b are positive constants, can be solved using separation of variables method.

Steady-State Solution

To find the steady-state solution, we set dy/dx = 0. This gives us y = a/b. Therefore, the steady-state solution is y = a/b.

Analysis of the Solution as x tends to Infinity

As x tends to infinity, the solution y(x) will tend towards the steady-state solution y = a/b. This is because as x becomes very large, the effect of the initial condition y(0) = ynot diminishes and the solution approaches the steady-state solution which is independent of the initial condition.

Explanation of the Asymptotic Behavior

The term -by^2 in the differential equation represents a negative feedback mechanism that pulls the solution towards the steady-state solution y = a/b as x tends to infinity. This negative feedback counteracts the growth term ay, causing the solution to stabilize at the steady-state solution.

Therefore, as x tends to infinity, the solution y(x) will approach the steady-state solution y = a/b. This behavior is a result of the interplay between the growth term and the negative feedback term in the differential equation.