Solve the differential equation.

(a) A direction field for the differential equation y0.3 is shown. Sketch the graphs of the solutions that satisfy the given initial conditions. (i) (ii) (iii) (iv) (b) If the initial condition is , for what values of is finite? What are the equilibrium solutions?

(a) Sketch a direction field for the differential equation . Then use it to sketch the four solutions that satisfy the initial conditions , , , and . (b) Check your work in part (a) by solving the differential equation explicitly. What type of curve is each solution curve?

(a) A direction field for the differential equation is shown. Sketch the solution of the initial-value problem Use your graph to estimate the value of . 0 x y _2 _1 1 2 1 2 _1 _2 _3 3 3 _3 y0.3 (b) Use Eulers method with step size 0.1 to estimate , where is the solution of the initial-value problem in part (a). Compare with your estimate from part (a). (c) On what lines are the centers of the horizontal line segments of the direction field in part (a) located? What happens when a solution curve crosses these lines?

(a) Use Eulers method with step size 0.2 to estimate , where is the solution of the initial-value problem (b) Repeat part (a) with step size 0.1. (c) Find the exact solution of the differential equation and compare the value at 0.4 with the approximations in parts (a) and (b).

Solve the differential equation.

Solve the differential equation.

Solve the initial-value problem

Solve the initial-value problem

Find the orthogonal trajectories of the family of curves.

Find the orthogonal trajectories of the family of curves.

A bacteria culture contains 200 cells initially and grows at a rate proportional to its size. After half an hour the population has increased to 360 cells. (a) Find the number of bacteria after hours. (b) Find the number of bacteria after 4 hours. (c) Find the rate of growth after 4 hours. (d) When will the population reach 10,000?

Cobalt-60 has a half-life of 5.24 years. (a) Find the mass that remains from a 100-mg sample after 20 years. (b) How long would it take for the mass to decay to 1 mg?

Let be the concentration of a drug in the bloodstream. As the body eliminates the drug, decreases at a rate that is proportional to the amount of the drug that is present at the time. Thus , where is a positive number called the elimination constant of the drug. (a) If is the concentration at time , find the concentration at time . (b) If the body eliminates half the drug in 30 hours, how long does it take to eliminate 90% of the drug?

A cup of hot chocolate has temperature in a room kept at . After half an hour the hot chocolate cools to . (a) What is the temperature of the chocolate after another half hour? (b) When will the chocolate have cooled to ?

(a) Write the solution of the initial-value problem and use it to find the population when . (b) When does the population reach 1200?

(a) The population of the world was 5.28 billion in 1990 and 6.07 billion in 2000. Find an exponential model for these data and use the model to predict the world population in the year 2020. (b) According to the model in part (a), when will the world population exceed 10 billion? (c) Use the data in part (a) to find a logistic model for the population. Assume a carrying capacity of 100 billion. Then use the logistic model to predict the population in 2020. Compare with your prediction from the exponential model. (d) According to the logistic model, when will the world population exceed 10 billion? Compare with your prediction in part (b).

The von Bertalanffy growth model is used to predict the length of a fish over a period of time. If is the largest length for a species, then the hypothesis is that the rate of growth in length is proportional to , the length yet to be achieved. (a) Formulate and solve a differential equation to find an expression for . (b) For the North Sea haddock it has been determined that , cm, and the constant of proportionality is . What does the expression for become with these data?

The Brentano-Stevens Law in psychology models the way that a subject reacts to a stimulus. It states that if represents the reaction to an amount of stimulus, then the relative rates of increase are proportional: where is a positive constant. Find as a function of .

One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of 5000 inhabitants, 160 people have a disease at the beginning of the week and 1200 have it at the end of the week. How long does it take for of the population to become infected?

A tank contains 100 L of pure water. Brine that contains 0.1 kg of salt per liter enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after 6 minutes?

The transport of a substance across a capillary wall in lung physiology has been modeled by the differential equation where is the hormone concentration in the bloodstream, is time, is the maximum transport rate, is the volume of the capillary, and is a positive constant that measures the affinity between the hormones and the enzymes that assist the process. Solve this differential equation to find a relationship between and .

Populations of birds and insects are modeled by the equations (a) Which of the variables, or , represents the bird population and which represents the insect population? Explain. (b) Find the equilibrium solutions and explain their significance. (c) Find an expression for . (d) The direction field for the differential equation in part (c) is shown. Use it to sketch the phase trajectory corresponding to initial populations of 100 birds and 40,000 insects. Then use the phase trajectory to describe how both populations change. (e) Use part (d) to make rough sketches of the bird and insect populations as functions of time. How are these graphs related to each other?

Suppose the model of Exercise 22 is replaced by the equations (a) According to these equations, what happens to the insect population in the absence of birds? (b) Find the equilibrium solutions and explain their significance. (c) The figure at the right shows the phase trajectory that starts with 100 birds and 40,000 insects. Describe what eventually happens to the bird and insect populations. (d) Sketch graphs of the bird and insect populations as functions of time.

Barbara weighs 60 kg and is on a diet of 1600 calories per day, of which 850 are used automatically by basal metabolism. She spends about 15 calkgday times her weight doing exercise. If 1 kg of fat contains 10,000 cal and we assume that the storage of calories in the form of fat is efficient, formulate a differential equation and solve it to find her weight as a function of time. Does her weight ultimately approach an equilibrium weight?