Exam 2: First-Order Differential Equations

The amount of medicine in the bloodstream decays exponentially with a half-life of 4 hours. There must be at least 25 milligrams of medicine per pound of body weight to keep the patient safe during a one-hour procedure. Set up an initial value problem whose solution is the amount of medicine, x(t) (measured in milligrams), in the bloodstream at time t. Let t = 0 correspond to the time when the procedure begins, and assume that x(0) = x₀. Determine the exact value of proportionality constant in the differential equation from.

Consider the autonomous differential equation Which of these is an equilibrium solution of this differential equation? Select all that apply.

Which of these differential equations models the following situation?In a town with 4 million people, the rate at which the inhabitants hear a rumor is proportional to the number of people who have not heard the rumor. Use N(t) for the number of people (in millions) who have heard the rumor at time t.

Consider the autonomous differential equation Determine for the initial condition Enter the exact answer.

Consider the initial value problem What can be said about the applicability of the existence and uniqueness theorem to this initial value problem?

Which of the following is the general solution of the differential equation

Consider the differential equation Which of the following statements regarding this differential equation is true?

Consider the autonomous differential equation Which of the following statements are true? Select all that apply.

A city's water reservoir contains 7 billion cubic meters (bcm) of water. The purification system ensures that the concentration of pollutants remains constant at 0.3 kilograms per bcm, and sensors will trigger an alarm if the concentration of pollutants rises above 1 kilogram per bcm. Water flows in and out of the reservoir at the same rate of 0.20 bcm per day, and the concentration of pollutants in the inflow is 1.9 kilograms per bcm. At all times, the reservoir is well mixed.Set up a differential equation whose solution x(t) is the amount of pollutant in the reservoir at time t and find the general solution of the equation.x(t) = _________________

A ball is thrown from the top of a tall building with a speed of 14 meters per second. Suppose the ball hits the ground with a speed of 69 meters per second. (Recall that the acceleration due to gravity is 9.8 m/s².)How tall (in meters) is the building? Round your answer to the nearest hundredth of a meter.

For each differential equation, select each category in which it falls.

Which of the following first-order differential equations are linear in y? Select all that apply.

Consider the difference equation Which of the following is an accurate description of the behavior of the solutions to this difference equation in terms of y₀?