Question

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?

(a) ( \frac{dp}{dt}=200 )
(b) ( \frac{dp}{dt}=200t )
(c) ( \frac{dp}{dt}=100t^{2} )
(d) ( \frac{dp}{dt}=200p )
(e) ( \frac{dp}{dt}=100p^{2} )

1. 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) )?

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