Assume that in the absence of immigration and emigration, the growth of a country's population $P(t)$ satisfies $\frac{dP}{dt} = kP$ for some constant $k > 0$ (see Equation (1) of Section 1.3). Determine a differential equation governing the growing population $P(t)$ of the country when individuals are allowed to immigrate into the country at a constant rate $r > 0$ . (Use $P$ for $P(t)$ .)

What is the differential equation for the population $P(t)$ of the country when individuals are allowed to emigrate at a constant rate $r > 0$ ?

Question :

Assume that in the absence of immigration and emigration, the growth of a country's population $p(t)$ satisfies $\frac{dp}{dt} = kp$ for some constant $k > 0$ (see equation (1) of section 1.3). determine a differential equation governing the growing population $p(t)$ of the country when individuals are allowed to immigrate into the country at a constant rate $r > 0$ . (use $p$ for $p(t)$ .)

what is the differential equation for the population $p(t)$ of the country when individuals are allowed to emigrate at a constant rate $r > 0$ ?

Assume that in the absence of immigration and emigration, the growth of a countr | Doubtlet.com

Solution:

Neetesh Kumar | November 08, 2024

Differential Equation Homework Help

This is the solution to Math 2A, section 13Z, Fall 2023 | WebAssign
Math002ACh1Sec03 (Homework) Question - 1
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Step-by-Step-Solution:

When individuals are allowed to immigrate into the country at a constant rate $r > 0$ , the rate of change of the population can be expressed as:

$\frac{dP}{dt} = kP + r$

This equation states that the growth rate of the population is due to both the natural growth of the population (given by $kP$ ) and the constant influx of new individuals (given by $r$ ).

Conclusion

The differential equation for the population $P(t)$ of the country when individuals are allowed to immigrate at a constant rate $r > 0$ is:

$\frac{dP}{dt} = \boxed{kP + r}$

What is the differential equation for the population $P(t)$ of the country when individuals are allowed to emigrate at a constant rate $r > 0$ ?

New Differential Equation

If individuals are allowed to emigrate at a constant rate $r > 0$ , then the rate of change of the population would decrease by that same rate. Therefore, the differential equation becomes:

$\frac{dP}{dt} = kP - r$

This represents the growth of the population minus the number of individuals leaving the country.

Conclusion

The differential equation for the population $P(t)$ of the country when individuals are allowed to emigrate at a constant rate $r > 0$ is:

$\frac{dP}{dt} = \boxed{kP - r}$

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