2. Consider the following differential equation:

Which of the following is true about a solution which satisfies the initial condition
?
(a) for all
(b) for all
© for all
(d) for all

If we consider the differential equation , we can analyze the behavior of the solutions based on the initial condition . When , the right-hand side of the equation becomes zero, which means that at this point, the solution will not change over time.

So, the only option that holds true based on this analysis is:
(b) for all .

Now, if we think about some real-world applications, this kind of differential equation could model a population or resource that stabilizes at a certain level when reaching a threshold, say, the population of a species in a controlled environment. When the initial population starts at a stable maximum (like 4), it simply remains that way unless external factors intervene.

Furthermore, a common mistake in solving such differential equations is failing to recognize the equilibrium points where . Remember, if your initial condition starts at an equilibrium point (in this case, at 4), the solution remains constant, highlighting the importance of where initial conditions place the solution.