PROBLEM SET #1. AVAILABLE 8/28; DUE 9/19 at 5:00 PM in Cu I Room 100.

Be sure to show your work.

(Comment on notation : "m^n" means "m to the power n".)

1. Draw a direction field for the differential equation y' = y - y^2. Identify the isoclines and any equilibrium solutions.
If there are equilibrium solutions, are they stable or unstable?

2. Same as problem 1, but now consider the equation y' = y^2 - y.

3. Find the general solution of y' + y = t^2. How do the solutions y(t) behave as t --> infinity?

4. Equations of the form y' = ay - by^3, where a and b are positive physical constants, arise in the study of fluid dynamics.
Solve the I.V.P. y' = y - y^3, y(0) = 1.

5. Solve the I.V.P. dy/dx = (1 - x)y^2, y(0) = 2. The solution is defined on an interval (-infinity, a). Find a.

6. (Baseball physics) A model for the velocity of a batted baseball is: dv/dt = -rv, dw/dt = -g - rw,
where r is the coefficient of air friction, g the acceleration due to gravity, and v and w are
respectively the x - and y - components of the velocity. Use this model, with r = 1/5 and g = 32 ft/sec^2 to solve the following problem:
A baseball is hit straight toward the left field fence with an initial velocity of 150 ft/sec, inclined at an angle of 30 degrees above
the horizontal. When hit, the ball is 3 feet above the ground. The left field fence is 10 feet tall, and lies 350 feet from the plate.
Does the ball clear the fence for a home run?

7. Refer to problem #6 in section 2.3 of the text and use Toricelli's principle to answer the following: A water tank has the form of a right circular cylinder with a radius of 1 meter and a height of 5 meters. In the center of the bottom is a circular hole with a radius of 1 centimeter. If the tank is filled to the top with water
and then allowed to drain, how long does it take the tank to empty?

8. (Refer to problem 19 in section 2.5.) Suppose in the situation of problem 7, water also enters the tank from a pipe at the rate of k m^3/sec. Is there an
equilibrium depth to the water? If so, express it as afunction of k and classify it as stable or unstable.

9. Solve the I.V.P. dy/dx = e^2x + y - 1, y(0) = 2.

10. Use the variable change u = ln(y) to solve the IVP dy/dt = -y ln(y), y(1) = 2.
(With some parameters added, equations like this are used in population modeling.)