Solving Differential Equations Word Problems and Initial-Value Problems

1. Find a general solution of the differntial equation
Then find a particular solution that satisfies the intiial condition .

2. A bacteria population is incresing according to the natural growth formula and numbers 100 at 12 noon and 156 at 1 p.m.
Write a formula giving after hours.

3. Apply Euler's method to the initial value problem below so as to approximate its solution on the interval

Use step size

4. Solve the initial value problem

5. Suppose Anytown, USA has a fixed population of 200,000. On March 1, 3000 people have the flu. On June 1, 6000 people have it.
If the rate of increase of the number N(t) who have the flu is proportional the number who don't have it, how many will have the disease on September 1?

6. Find the particular solution of the differential equation subject to the initial condition .

7. Suppose that a motorboat is moving at 30 ft/sec when its motor suddenly quits, and that 5 seconds later the boat has slowed to 15 ft/sec. Assume that the resistance it encounters while coasting is porportional to its velocity. How far will the boat coast in all?

8. The time rate of change of an alligator population P in a swamp is proportional to the square root of P. The swamp contained 9 alligators in 1990 and 25 alligators in 1995. When will there be 49 alligators in the swamp?

9. Find teh general solution of the differential equation:

10. Solve the initial value problem:

11. The solution describes forced undamped motion of mass on a spring. Express the position function x(t) as the sum of two oscillations.

12. A mass of 25g is attached to a vertical spring with a spring constant dyne/cm. The surrounding medium has a damping constant of 10 dyne*sec/cm. The mass is pushed 5 cm above its equlibrium position and released. Find (a) the position function of the mass, (b) the period of the vibration, and (c) the frequency of the vibration.

Please see the attached file for the fully formatted problems.