Deck 9: Differential Equations

Question 1

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.
​ , ​ , __________
​ , __________

Answer 1

24.1367; 30.1719

Question 2

It has been estimated that service industries, which currently make up 27% of the nonfarm workforce in a certain country, will continue to grow at the rate of​ ​ percent per decade t decades from now. Estimate the percentage of the nonfarm workforce in the service industries one decade from now. Round your answer to the nearest integer.

A) Show that the desired answer is P(1), where P is the solution of the initial value problem
B) Use Euler's method with n=10 to approximate the solution. P(1)= ________ %

Answer 2

A)
B) 50

Question 3

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.
​ , ​ , __________
​ , __________

Answer 3

2.1045; 2.1497

Question 4

The proprietor of Carson Hardware Store has decided to set up a sinking fund for the purpose of purchasing a computer 3 yr from now. It is expected that the purchase will involve a sum of $30,000. The fund grows at the rate of where A denotes the size of the fund at any time t, r is the annual interest rate earned by the fund compounded continuously, and P is the amount (in dollars) paid into the fund by the proprietor per year (assume this is done on a frequent basis in small deposits over the year so that it is essentially continuous). If the fund earns 8% interest per year compounded continuously, determine the size of the yearly investment the proprietor should pay into the fund. Round the answer to the nearest dollar. $__________

Question 5

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

Question 6

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.
​ , ​ , __________
​ , __________

Question 7

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

Question 8

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

Question 9

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.
​ , ​ , __________
​ , __________

Question 10

Under ideal laboratory conditions, the rate of growth of bacteria in a culture is proportional to the size of the culture at any time t. Suppose that 4,000 bacteria are present initially in the culture and 5,000 are present 1 hr later. How many bacteria will be in the culture at the end of 4 hr? Round the answer to the nearest thousand. __________ bacteria

Question 11

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)

Question 12

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.
​ , ​ , __________
​ , __________

Question 13

A biologist has determined that the maximum number of fruit flies that can be sustained in a carefully controlled environment (with a limited supply of space and food) is 500. Suppose that the rate at which the population of the colony increases obeys the rule where C is the carrying capacity (500) and Q denotes the number of fruit flies in the colony at any time t. If the initial population of fruit flies in the experiment is 20 and it grows to 35 after 20 days, determine the population of the colony of fruit flies on the 50th day. Round the answer to the nearest integer. __________ flies

Question 14

Newton's law of cooling states that the rate at which the temperature of an object changes is directly proportional to the difference in temperature between the object and that of the surrounding medium. A cup of coffee is prepared with boiling water (212 °F) and left to cool on the counter in a room where the temperature is 70 °F. If the temperature of the coffee is 120 °F after 1 min, determine when the coffee will be cool enough to drink (say, 111 °F). Round the answer to the nearest tenth, if necessary. t = __________ min

Question 15

​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.
​ , ​ , __________
​ , __________

Question 16

The Institute finds that the average student taking Elementary Shorthand will progress at a rate given by ​ in a 30-wk course, where Q(t) measures the number of words of dictation a student can take per minute after t wk in the course. If the average student can take 40 words of dictation per minute after 15 wk in the course, how many words per minute can the average student take after completing the course? Round the answer to the nearest word. __________ words

Question 17

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.
​ , ​ , ​ __________
​ , ​ __________

Question 18

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.
​ , ​ , __________
​ , __________

Question 19

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.
​ , ​ , __________
​ , __________

Question 20

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.
​ , ​ , __________
​ , __________

Question 21

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

Question 22

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

Question 23

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

Question 24

Find the solution of the initial value problem.
​

Question 25

An amount of money deposited in a savings account grows at a rate proportional to the amount present. (It can be shown that an amount of money grows in this manner if it earns interest compounded continuously.) Suppose $50,000 is deposited in a fixed account earning interest at the rate of 8%/year compounded continuously. What is the accumulated amount after 6 yr? Round the answer to the nearest cent. $__________ How long does it take for the original deposit to double in value? Round the answer to the nearest tenth, if necessary. __________ yr

Question 26

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.

Question 27

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

Question 28

A tank initially contains 60 gal of brine, in which 10 lb of salt is dissolved. Brine containing 2 lb of dissolved salt per gallon flows into the tank at the rate of 4 gal/min, and the well-stirred mixture flows out of the tank at the same rate. How much salt is present in the tank at the end of 10 min?Round the answer to the nearest tenth, if necessary.__________ lb

Question 29

Find the solution of the initial value problem.
​

Question 30

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

Question 31

Find the solution of the initial value problem.
​

Question 32

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

Question 33

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

Question 34

According to Lambert's law of absorption, the percentage of incident light L, absorbed in passing through a thin layer of material x, is proportional to the thickness of the material. If in. of a certain material reduces the light to half of its intensity, how much additional material is needed to reduce the intensity to one-fourth of its initial value?

Question 35

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. If the total population of the country at time t = 0 is P​₀, find an expression for the population at any time t.

Question 36

Find the solution of the initial value problem.
​

Question 37

Find the solution of the initial value problem.
​

Question 38

The rate of decomposition of a certain chemical substance is directly proportional to the amount present at any time t. If d₀ gr of the chemical are present at time t = 0, find an expression for the amount present d(t) at any time t.

Question 39

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

Question 40

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

Question 41

Assume that the rate of change of the unit price of a commodity is proportional to the difference between the demand and the supply so that ​ where k is a constant of proportionality. Suppose that D=74-5p, S=2+3p, and p(0)=2. Find a formula for p(t).

Question 42

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 .

Question 43

y is a solution of the differential equation.
​ ​

Question 44

The rate at which the concentration of a drug in the bloodstream decreases is proportional to the concentration at any time t. Initially, the concentration of the drug in the bloodstream is B₀ g/mL. What is the concentration of the drug in the bloodstream at any time t? Formulate but do not solve the problem in terms of a differential equation with a side condition.

Question 45

A radioactive substance decays at a rate directly proportional to the amount present. If the substance is present in the amount of g initially , find the amount present in grams at any time . Formulate but do not solve the problem in terms of a differential equation with a side condition.

Question 46

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

Question 47

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 .

Question 48

Find the solution of the initial value problem.
​

Question 49

Find the solution of the initial value problem.
​

Question 50

Choose a solution of the differential equation. ​ ​

A) ​
B) ​
C) ​
D) ​

Question 51

y is a solution of the differential equation.n ​ any constant; ​

Question 52

The management of a company has decided that the level of investment should not exceed D dollars. Furthermore, management has decided that the rate of net investment (the rate of change of the total capital invested) should be proportional to the difference between D and the total capital invested (A). Formulate but do not solve the problem in terms of a differential equation.

Question 53

y is a solution of the differential equation.
​
​

Question 54

Find the solution of the initial value problem.
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Question 55

Find a function f given that the slope of the tangent line to the graph of f at any point P(x, y) is given by the expression and the graph of f passes through the point (0, 2).

Question 56

Find the solution of the initial value problem.
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Question 57

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