Biology A population of eight beavers has been introduced into a new wetlands area

In Exercises 112, find the general solution of the differential equation.

In Exercises 112, find the general solution of the differential equation.

In Exercises 112, find the general solution of the differential equation.

In Exercises 112, find the general solution of the differential equation.

In Exercises 112, find the general solution of the differential equation.

In Exercises 112, find the general solution of the differential equation.

In Exercises 112, find the general solution of the differential equation.

In Exercises 112, find the general solution of the differential equation.

In Exercises 112, find the general solution of the differential equation.

In Exercises 112, find the general solution of the differential equation.

In Exercises 112, find the general solution of the differential equation.

In Exercises 112, find the general solution of the differential equation.

In Exercises 1322, find the particular solution that satisfies the initial condition.yy  e y0
4 x

In Exercises 1322, find the particular solution that satisfies the initial condition.
x
y y
0 y1
4 y

In Exercises 1322, find the particular solution that satisfies the initial condition.yx 1 y
0 y2
1
x

In Exercises 1322, find the particular solution that satisfies the initial condition.2xy  ln x y1
2 2

In Exercises 1322, find the particular solution that satisfies the initial condition.y1 x y0

3 2y

In Exercises 1322, find the particular solution that satisfies the initial condition.y
1  x y0
1 2

In Exercises 1322, find the particular solution that satisfies the initial condition.

In Exercises 1322, find the particular solution that satisfies the initial condition.dr ds
er2s

In Exercises 1322, find the particular solution that satisfies the initial condition.dP  P0
P0 kP dt
0

In Exercises 1322, find the particular solution that satisfies the initial condition.dT kT  70 dt
0 T0
140 dP

In Exercises 23 and 24, find an equation of the graph that passes through the point and has the given slope.

In Exercises 23 and 24, find an equation of the graph that passes through the point and has the given slope.

In Exercises 25 and 26, find all functions f having the indicated property.

In Exercises 25 and 26, find all functions f having the indicated property.

In Exercises 2734, determine whether the function is homogeneous, and if it is, determine its degree.fx, y
x3  4xy2 y3 f

In Exercises 2734, determine whether the function is homogeneous, and if it is, determine its degree.fx, y
x3 3x2y2  2y2 f

In Exercises 2734, determine whether the function is homogeneous, and if it is, determine its degree.fx, y
x2y2
x2 y2 f

In Exercises 2734, determine whether the function is homogeneous, and if it is, determine its degree.fx, y
xy
x2 y2 f

In Exercises 2734, determine whether the function is homogeneous, and if it is, determine its degree.fx, y
2 ln xy

In Exercises 2734, determine whether the function is homogeneous, and if it is, determine its degree.fx, y
tanx y fx

In Exercises 2734, determine whether the function is homogeneous, and if it is, determine its degree.fx, y
2 ln x

In Exercises 2734, determine whether the function is homogeneous, and if it is, determine its degree.fx, y
tan y x

In Exercises 35 40, solve the homogeneous differential equation.y
x y 2x

In Exercises 35 40, solve the homogeneous differential equation.y
x3 y3 xy2

In Exercises 35 40, solve the homogeneous differential equation.y
x  y x y

In Exercises 35 40, solve the homogeneous differential equation.y
x2 y2 2xy

In Exercises 35 40, solve the homogeneous differential equation.y
xy x2  y2

In Exercises 35 40, solve the homogeneous differential equation.y
2x 3y x

In Exercises 4144, find the particular solution that satisfies the initial condition.

In Exercises 4144, find the particular solution that satisfies the initial condition.

In Exercises 4144, find the particular solution that satisfies the initial condition.

In Exercises 4144, find the particular solution that satisfies the initial condition.

Slope Fields In Exercises 4548, sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. To print an enlarged copy of the graph, go to the website www.mathgraphs.com

Slope Fields In Exercises 4548, sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. To print an enlarged copy of the graph, go to the website www.mathgraphs.com

Slope Fields In Exercises 4548, sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. To print an enlarged copy of the graph, go to the website www.mathgraphs.com

Slope Fields In Exercises 4548, sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. To print an enlarged copy of the graph, go to the website www.mathgraphs.com

Eulers Method In Exercises 4952, (a) use Eulers Method with a step size of to approximate the particular solution of the initial value problem at the given -value, (b) find the exact solution of the differential equation analytically, and (c) compare the solutions at the given -value.

Eulers Method In Exercises 4952, (a) use Eulers Method with a step size of to approximate the particular solution of the initial value problem at the given -value, (b) find the exact solution of the differential equation analytically, and (c) compare the solutions at the given -value.

Eulers Method In Exercises 4952, (a) use Eulers Method with a step size of to approximate the particular solution of the initial value problem at the given -value, (b) find the exact solution of the differential equation analytically, and (c) compare the solutions at the given -value.

Eulers Method In Exercises 4952, (a) use Eulers Method with a step size of to approximate the particular solution of the initial value problem at the given -value, (b) find the exact solution of the differential equation analytically, and (c) compare the solutions at the given -value.

Radioactive Decay The rate of decomposition of radioactive radium is proportional to the amount present at any time. The half-life of radioactive radium is 1599 years. What percent of a present amount will remain after 25 years?

Chemical Reaction In a chemical reaction, a certain compound changes into another compound at a rate proportional to the unchanged amount. If initially there is 20 grams of the original compound, and there is 16 grams after 1 hour, when will 75 percent of the compound be changed?

Slope Fields In Exercises 5558, (a) write a differential equation for the statement, (b) match the differential equation with a possible slope field, and (c) verify your result by using a graphing utility to graph a slope field for the differential equation. [The slope fields are labeled (a), (b), (c), and (d).] To print an enlarged copy of the graph, go to the website www.mathgraphs.com.The rate of change of with respect to is proportional to the difference between and 4.

Slope Fields In Exercises 5558, (a) write a differential equation for the statement, (b) match the differential equation with a possible slope field, and (c) verify your result by using a graphing utility to graph a slope field for the differential equation. [The slope fields are labeled (a), (b), (c), and (d).] To print an enlarged copy of the graph, go to the website www.mathgraphs.com.The rate of change of with respect to is proportional to the difference between and 4.

Slope Fields In Exercises 5558, (a) write a differential equation for the statement, (b) match the differential equation with a possible slope field, and (c) verify your result by using a graphing utility to graph a slope field for the differential equation. [The slope fields are labeled (a), (b), (c), and (d).] To print an enlarged copy of the graph, go to the website www.mathgraphs.com.The rate of change of with respect to is proportional to the product of and the difference between and 4.

Slope Fields In Exercises 5558, (a) write a differential equation for the statement, (b) match the differential equation with a possible slope field, and (c) verify your result by using a graphing utility to graph a slope field for the differential equation. [The slope fields are labeled (a), (b), (c), and (d).] To print an enlarged copy of the graph, go to the website www.mathgraphs.com.The rate of change of with respect to is proportional to y2

Weight Gain A calf that weighs 60 pounds at birth gains weight at the rate where is weight in pounds and is time in years. Solve the differential equation. (a) Use a computer algebra system to solve the differential equation for and 1. Graph the three solutions. (b) If the animal is sold when its weight reaches 800 pounds, find the time of sale for each of the models in part (a). (c) What is the maximum weight of the animal for each of the models?

Weight Gain A calf that weighs pounds at birth gains weight at the rate where is weight in pounds and is time in years. Solve the differential equation.

In Exercises 61 66, find the orthogonal trajectories of the family. Use a graphing utility to graph several members of each family.x2 y2
C

In Exercises 61 66, find the orthogonal trajectories of the family. Use a graphing utility to graph several members of each family.x2  2y2
C

In Exercises 61 66, find the orthogonal trajectories of the family. Use a graphing utility to graph several members of each family.x2
Cy

In Exercises 61 66, find the orthogonal trajectories of the family. Use a graphing utility to graph several members of each family.y2
2Cx

In Exercises 61 66, find the orthogonal trajectories of the family. Use a graphing utility to graph several members of each family.y2
Cx3

In Exercises 61 66, find the orthogonal trajectories of the family. Use a graphing utility to graph several members of each family.y
Cex

Biology At any time t, the rate of growth of the population N of deer in a state park is proportional to the product of N and where is the maximum number of deer the park can sustain. When and when N
200. Write N as a function of t. t
0, N

Sales Growth The rate of change in sales S (in thousands of units) of a new product is proportional to the product of S and L (in thousands of units) is the estimated maximum level of sales, and when Write and solve the differential equation for this sales model.

Chemical Reaction In Exercises 69 and 70, use the chemical reaction model given in Example 10 to find the amount y as a function of t, and use a graphing utility to graph the function.

Chemical Reaction In Exercises 69 and 70, use the chemical reaction model given in Example 10 to find the amount y as a function of t, and use a graphing utility to graph the function.

In Exercises 71 and 72, use the Gompertz growth model described in Example 11 to find the growth function, and sketch its graph.

In Exercises 71 and 72, use the Gompertz growth model described in Example 11 to find the growth function, and sketch its graph.

Biology A population of eight beavers has been introduced into a new wetlands area. Biologists estimate that the maximum population the wetlands can sustain is 60 beavers. After 3 years, the population is 15 beavers. If the population follows a Gompertz growth model, how many beavers will be present in the wetlands after 10 years?

Biology A population of 30 rabbits has been introduced into a new region. It is estimated that the maximum population the region can sustain is 400 rabbits. After 1 year, the population is estimated to be 90 rabbits. If the population follows a Gompertz growth model, how many rabbits will be present after 3 years?

Biology In Exercises 75 and 76, use the hybrid selection model described in Example 12 to find the percent of the population that has the indicated characteristic

Biology In Exercises 75 and 76, use the hybrid selection model described in Example 12 to find the percent of the population that has the indicated characteristic

Chemical Mixture A 100-gallon tank is full of a solution containing 25 pounds of a concentrate. Starting at time distilled water is admitted to the tank at the rate of 5 gallons per minute, and the well-stirred solution is withdrawn at the same rate, as shown in the figure.(a) Find the amount Q of the concentrate in the solution as a function of t. (Hint: ) (b) Find the time when the amount of concentrate in the tank reaches 15 pounds

Chemical Mixture A 200-gallon tank is half full of distilled water. At time a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 5 gallons per minute, and the well-stirred mixture is withdrawn at the same rate. Find the amount Q of concentrate in the tank after 30 minutes

Chemical Reaction In a chemical reaction, a compound changes into another compound at a rate proportional to the unchanged amount, according to the model (a) Solve the differential equation. (b) If the initial amount of the original compound is 20 grams, and the amount remaining after 1 hour is 16 grams, when will 75% of the compound have been changed?

Safety Assume that the rate of change in the number of miles s of road cleared per hour by a snowplow is inversely proportional to the depth h of snow. That is, Find s as a function of h if miles when inches and miles when inches 2 h 15.

Chemistry A wet towel hung from a clothesline to dry loses moisture through evaporation at a rate proportional to its moisture content. If after 1 hour the towel has lost 40% of its original moisture content, after how long will it have lost 80%?

Biology Let x and y be the sizes of two internal organs of a particular mammal at time t. Empirical data indicate that the relative growth rates of these two organs are equal, and can be modeled by Use this differential equation to write y as a function of x.

Population Growth When predicting population growth, demographers must consider birth and death rates as well as the net change caused by the difference between the rates of immigration and emigration. Let P be the population at time t and let N be the net increase per unit time due to the difference between immigration and emigration. So, the rate of growth of the population is given by N is constant. Solve this differential equation to find P as a function of time.

Meteorology The barometric pressure y (in inches of mercury) at an altitude of x miles above sea level decreases at a rate proportional to the current pressure according to the model where inches when Find the barometric pressure (a) at the top of Mt. St. Helens (8364 feet) and (b) at the top of Mt. McKinley (20,320 feet).

Investment A large corporation starts at time to invest part of its receipts at a rate of P dollars per year in a fund for future corporate expansion. Assume that the fund earns r percent interest per year compounded continuously. So, the rate of growth of the amount A in the fund is given by where when Solve this differential equation for A as a function of t.

Investment In Exercises 8688, use the result of Exercise 85.

Investment In Exercises 8688, use the result of Exercise 85.Find P if the corporation needs $120,000,000 in 8 years and the fund earns interest compounded continuously

Investment In Exercises 8688, use the result of Exercise 85.Find t if the corporation needs $800,000 and it can invest $75,000 per year in a fund earning 8% interest compounded continuously

In Exercises 89 and 90, use the Gompertz growth model described in Example 11.

In Exercises 89 and 90, use the Gompertz growth model described in Example 11.

In your own words, describe how to recognize and solve differential equations that can be solved by separation of variables.

State the test for determining if a differential equation is homogeneous. Give an example.

In your own words, describe the relationship between two families of curves that are mutually orthogonal.

Sailing Ignoring resistance, a sailboat starting from rest accelerates at a rate proportional to the difference between the velocities of the wind and the boat. (a) The wind is blowing at 20 knots, and after 1 minute the boat is moving at 5 knots. Write the velocity as a function of time (b) Use the result of part (a) to write the distance traveled by the boat as a function of time.

True or False? In Exercises 95 98, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

True or False? In Exercises 95 98, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

True or False? In Exercises 95 98, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

True or False? In Exercises 95 98, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

A not uncommon calculus mistake is to believe that the product rule for derivatives says that If determine, with proof, whether there exists an open interval and a nonzero function defined on such that this wrong product rule is true for in (a, b).