Maths C Assignment
Table of Contents
Model A
The very first differential equation that one typically encounters is the equation that models the change of a population as being proportional to the number of individuals in the population. In symbols, if P(t) represents the number of individuals in a population at time , then the so called called exponential growth model is:
Recall that the general solution of this differential equation is . Recall also that in order to get a particular solution, we must have some sort of experimental observations that tell us:
• the initial population
• the net birth rate.
The sign of k determines whether the population will grow without limit, or whether it will become extinct. Question 1:
Using the data given in the table, solve the differential equation , to find a population function, P(t), for the USA. Show all working.
Solving Differential Equation:
When
in the year 1790 because this is the population when
To find in the equation for the population function using the data from the year 1800, the values of and are substituted in: the population of the USA in the year 1800 which is 5.5 million. the population of the USA in 1790 which is 3.6 million. the number of years since 1790 which is 10. constant rate of population growth.
Therefore the population function using the data from the year 1800 is
To find in the equation for the population function using the data from the year 1880, the values of and are substituted in: the population of the USA in 1880 which is 50.2 million the population of the USA in 1790 which is 3.6 million. the number of years since 1790 which is 90. constant rate of population growth.
Therefore the population function using
Model A
The very first differential equation that one typically encounters is the equation that models the change of a population as being proportional to the number of individuals in the population. In symbols, if P(t) represents the number of individuals in a population at time , then the so called called exponential growth model is:
Recall that the general solution of this differential equation is . Recall also that in order to get a particular solution, we must have some sort of experimental observations that tell us:
• the initial population
• the net birth rate.
The sign of k determines whether the population will grow without limit, or whether it will become extinct. Question 1:
Using the data given in the table, solve the differential equation , to find a population function, P(t), for the USA. Show all working.
Solving Differential Equation:
When
in the year 1790 because this is the population when
To find in the equation for the population function using the data from the year 1800, the values of and are substituted in: the population of the USA in the year 1800 which is 5.5 million. the population of the USA in 1790 which is 3.6 million. the number of years since 1790 which is 10. constant rate of population growth.
Therefore the population function using the data from the year 1800 is
To find in the equation for the population function using the data from the year 1880, the values of and are substituted in: the population of the USA in 1880 which is 50.2 million the population of the USA in 1790 which is 3.6 million. the number of years since 1790 which is 90. constant rate of population growth.
Therefore the population function using