Exam 9: Differential Equations

The population of a certain community is increasing at a rate directly proportional to the population at any time t. In the last 1 yr, the population has doubled. How long will it take for the population to triple? Round the answer to the nearest hundredth, if necessary. __________ yr(s)

y​ is a solution of the differential equation. ​ ​

The rate at which a rumor spreads through an Alpine village of 500 residents is jointly proportional to the number of residents who have heard it and the number who have not. Initially, 15 residents heard the rumor, but 4 days later this number increased to 80. Find the number of people who will have heard the rumor after 1 wk. Round the answer to the nearest integer. __________ people

Consider the differential equation ​ with the side condition . The solution Q(t) describes restricted growth and has a graph known as the Gompertz curve. Using separation of variables, solve this differential equation.

Find the solution of the initial value problem. ​

Two chemical solutions, one containing N molecules of chemical A and another containing M molecules of chemical B, are mixed together at time t = 0. The molecules from the two chemicals combine to form another chemical solution containing y (AB) molecules. The rate at which the AB molecules are formed, , is called the reaction rate and is jointly proportional to and . Thus, ​ where k is a constant (we assume the temperature of the chemical mixture remains constant during the interaction). Solve this differential equation with the side condition assuming that and

The personnel manager of Gibraltar Insurance Company estimates that the number of insurance claims an experienced clerk can process in a day is 50. Furthermore, the rate at which a clerk can process insurance claims during the t ^th wk of training is proportional to the difference between the maximum number possible (50) and the number he or she can process in the t ^th wk. If the number of claims the average trainee can process after 1 wk on the job is 20/day, determine how many claims the average trainee can process after 7 wk on the job. Round the answer to the nearest integer. __________ claims

​Use Euler's method with given values of n to obtain an approximation of the initial value problem when . Round your answers to four decimal places, if necessary. ​ , ​ , __________ ​ , __________

Find the general solution of the first-order differential equation by separating variables. ​

Find the general solution of the first-order differential equation by separating variables. ​

Use Euler's method with given values of n to obtain an approximation of the initial value problem when . Round your answers to four decimal places, if necessary. ​ , ​ , __________ ​ , __________

Suppose a country's population at any time t grows in accordance with the rule ​ where P denotes the population at any time t, k is a positive constant reflecting the natural growth rate of the population, and I is a constant giving the (constant) rate of immigration into the country. The population of the United States in the year 1980 was 224.5 million. Suppose the natural growth rate is 0.8% annually and net immigration is allowed at the rate of 0.5 million people/year until the end of the century. What will be the U.S. population in 2003? Round the answer to the nearest tenth of a million, if necessary. P=__________ million

Find the solution of the initial value problem. ​

Use Euler's method with given values of n to obtain an approximation of the initial value problem when . Round your answers to four decimal places, if necessary. ​ , ​ , __________ ​ , __________

Find the solution of the initial value problem. ​

y is a solution of the differential equation. ​ ​

Find the general solution of the first-order differential equation by separating variables. ​

Use Euler's method with given values of n to obtain an approximation of the initial value problem when . Round your answers to four decimal places, if necessary. ​ , ​ , __________ ​ , __________

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The function is a solution of the differential equation

Choose a solution of the differential equation. ​ ​