Question

Under the same assumptions that underline the model in (1), determine a differential equation for the population P(t) of a country when individuals are allowed to immigrate into the country at a constant rate r>0. What is the differential equation for the population P(t) of the country when individuals are allowed to emigrate from the country at a constant rate r>0?

Short answer

• Immigration into country $\frac{dP}{dt}=kP+r$
• Emigration from the country$\frac{dP}{dt}=kP-r$

Step-by-step solution

Definition of differential equation and formula  

A system of ordinary differential equations is two or more equations involving the derivatives of two or more unknown functions of a single independent variable.

$dy/dx=f\left(x\right)$

Determination of immigrant and emigrant

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

When there is a immigrate at a constant rate r>0, the population will be in greater rate.

Now we can add r to population model which implies population increased by the effect of immigration

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

When there is a emigrate at a constant rate r>0, the population rate is decreased.

Now we can subtract r to population model which implies population decreased by the effect of emigration

HenceImmigration into country$\frac{dP}{dt}=kP+r$and Emigration from the country$\frac{dP}{dt}=kP-r$