Exam 9: Differential Equations

A population is modeled by the differential equation . For what values of P is the population decreasing?

Suppose that a population grows according to a logistic model with carrying capacity and per year. Write the logistic differential equation for these data.

Which of the following functions is a solution of the differential equation?

A function satisfies the differential equation . What are the constant solutions of the equation?

Suppose that a population develops according to the logistic equation , where t is measured in weeks. What is the carrying capacity?

A sum of is invested at interest. If is the amount of the investment at time t for the case of continuous compounding, write a differential equation and an initial condition satisfied by .

Use Euler's method with step size 0.25 to estimate , where is the solution of the initial-value problem. Round your answer to four decimal places.

Determine whether the differential equation is linear.

Solve the initial-value problem.

A curve passes through the point and has the property that the slope of the curve at every point P is times the y-coordinate P. What is the equation of the curve?

Find the solution of the differential equation that satisfies the initial condition .

Find the solution of the initial-value problem and use it to find the population when .

Solve the initial-value problem.

A phase trajectory is shown for populations of rabbits and foxes . Describe how each population changes as time goes by. Select the correct statement.

For what nonzero values of k does the function satisfy the differential equation for all values of A and B?

The rate of change of atmospheric pressure P with respect to altitude h is proportional to P provided that the temperature is constant. At the pressure is at sea level and at . What is the pressure at an altitude of ?

A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every . The initial population of a culture is cells. Find the number of cells after hours.

Experiments show that if the chemical reaction takes place at , the rate of reaction of dinitrogen pentoxide is proportional to its concentration as follows : How long will the reaction take to reduce the concentration of to 50% of its original value?

Solve the initial-value problem.

The population of the world was about 5.3 billion in 1990. Birth rates in the 1990s range from 35 to 40 million per year and death rates range from 15 to 20 million per year. Let's assume that the carrying capacity for world population is 100 billion. Use the logistic model to predict the world population in the 2,450 year. Calculate your answer in billions to one decimal place. (Because the initial population is small compared to the carrying capacity, you can take k to be an estimate of the initial relative growth rate.)