1. if ( p(t) ) is the size of a population at time ( t ), which of the following differential equations…

1. if ( p(t) ) is the size of a population at time ( t ), which of the following differential equations describes linear growth in the size of the population?\n(a) ( \frac{dp}{dt}=200 )\n(b) ( \frac{dp}{dt}=200t )\n(c) ( \frac{dp}{dt}=100t^{2} )\n(d) ( \frac{dp}{dt}=200p )\n(e) ( \frac{dp}{dt}=100p^{2} )\n2. which of the following could be the graph of a solution of the differential equation ( \frac{dy}{dx}=(x - 1)y^{2} ) near the point ( (1,1) )?

1. if ( p(t) ) is the size of a population at time ( t ), which of the following differential equations describes linear growth in the size of the population?\n(a) ( \frac{dp}{dt}=200 )\n(b) ( \frac{dp}{dt}=200t )\n(c) ( \frac{dp}{dt}=100t^{2} )\n(d) ( \frac{dp}{dt}=200p )\n(e) ( \frac{dp}{dt}=100p^{2} )\n2. which of the following could be the graph of a solution of the differential equation ( \frac{dy}{dx}=(x - 1)y^{2} ) near the point ( (1,1) )?

Answer

Explanation:

Step1: Recall the form of linear growth

Linear growth means the rate of change of the population (P(t)) with respect to time (t), (\frac{dP}{dt}), is a constant.

Step2: Analyze each option

  • For option (A): (\frac{dP}{dt}=200), where the rate of change is a constant.
  • For option (B): (\frac{dP}{dt} = 200t), the rate of change is a linear function of (t) (not a constant).
  • For option (C): (\frac{dP}{dt}=100t^{2}), the rate of change is a quadratic function of (t) (not a constant).
  • For option (D): (\frac{dP}{dt}=200P), the rate of change is a function of (P) (exponential - like growth, not linear).
  • For option (E): (\frac{dP}{dt}=100P^{2}), the rate of change is a quadratic function of (P) (not linear).

Answer:

A. (\frac{dP}{dt}=200)