January 10, 2017

7.2 Exponential Growth and Decay

Differential Equations: Equations that display the rate of change of something.

For Example:

The rate of change of a population or dP/dt is directly proportional to that population. dP/dt = kP

Explain:

January 10, 2017

The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year. Write a differential equation that expresses this fact. Solve it to find an equation that expresses population as a function of time.

January 10, 2017

The number of fruit flies increase continuously at a rate proportional to the number present.

a. Write a differential equation to represent this situation using y for the number of fruit flies present.

b. Solve the differential equation to get a general equation for y.

c. If there were 100 flies after the 2nd day of the experiment and 300 flies after the 4th day, find the particular equation that represents the situation by substituting the new information into the equation found in part b.

d. How many flies were in the original population?

FRUIT FLY PROBLEM

January 10, 2017

Punctured Tire Problem:You have just run over a nail. As air leaks out of your tire, the rate of change of the air pressure inside the tire is directly proportional to that pressure.

a. Write a differential equation that states this fact. Evaluate the proportionality constant if, at time zero, the pressure is 35 psi and decreasing at 0.28 psi/min

b. solve the differential equation subject to the initial condition given in part a

c. Sketch the graph of the function

d. What is pressure 10 min after tire was punctured

e. The car is safe to drive as long as the tire pressure is 12 psi or greater. For how long after the puncture will the car be safe to drive.

January 10, 2017

Differential EquationsWrite a differential equation for each of the following statements, then solve for the general solution.

The rate of change of y with respect to x is directly proportional to x.

The rate of change of Q with respect to t is inversely proportional to the square of t.

January 10, 2017

The rate of change of P with respect to t is proportional to 10-t.

The rate of change of y with respect to x varies jointly as x and L-y.

January 10, 2017

1. 2.

3. 4.

5. 6.

January 10, 2017

7.3 Solving Differential Equations

7. Given

find the particular solution y = f(x) to the differential equation with the condition f(1) = 0.

January 10, 2017

8. Given , where . Find the particular solution y = f(x) to the differential equation with initial condition f(-1) = 1 and state its domain.

January 10, 2017

9. Given let y = f(x) be the particular solution to this differential equation with the initial condition f(-1) = 2. Write an equation for the line tangent to the graph of f at x=-1. Then find the solution y = f(x) to the given differential equation with the initial condition f(-1) = 2.

January 10, 2017

10. Given . Let y = f(x) be the particular solution to the differential equation with the initial condition f(1) = -1. Write an equation for the line tangent to the graph of f at (1, -1) and use it to approximate f(1.1). Then find the particular solution y = f(x) to the given differential equation with the initial condition f(1)=-1.

January 10, 2017

11. Given . Can we use separation of variable to solve for y = f(x)? Why or why not? The solution of the curve that passes through the point (0, 1) has a local minimum at .

What is the y-coordinate of this local minimum?

January 10, 2017

12. A population is modeled by the function Y that satisfies the separable differential equation Find Y(t) if Y(0) = 3. Then use your function to evaluate .