Fall 2005

**Practice Exam 3**

Kingston 11/15/2005

** **

**1. **
(a) Find the general solution to the
differential equation

**y ' = ( x ^{2} +1) y**

(b) Solve the initial value problem

**y ' = ( x ^{2} +1) y, y(0) = 2.**

**2. **

(a) ** **Find the general solution to the
differential equation

**y ' = ( 2 x +1) (y + 2).**

(b) Solve the initial value problem

**y ' = ( 2 x +1) (y + 2), y(0) = 1.**

**3. **A population of insects at time*
t, Q(t) * increases at a rate
proportional to the size of the population, with the coefficient of
proportionality

(a) Write a differential equation for the
function **Q(t).**

(b) Find the general solution to this equation.

(c) If the initial population is * 700*
, how
many insects will be present after

**4. ** A yam is put in a * 220 ^{o}
C *oven and heats up according to the Newton's Law of
Heating.

(a) If the yam is at * 22
^{o} C* when it is put in oven, write a differential equation
satisfied by the temperature,

(b) Solve the differential equation in (a)
using the fact after * 20* minutes the temperature
of the yam is

(c) Calculate the temperature of the yam after * 35*
minutes

**5**. The amount of a drug in a patient's
bloodstream, * Q(t)*, in

(a) Write a differential equation for the
function * Q(t)*.

(b) Find the general solution to the equation.

(c) If the initial amount * Q(0)=500*,
how much drug is left after

**6**. The following system of differential equations
represents the interaction between two populations, x and y.

**dx/dt = x + 0.5 x y**

**dx/dt = -y + 0.4 x y**

(a) Describe how the species interact. Are they helpful or harmful to each other ?

(b) Write a differential equation for * dy/dx*
.

**
7. ** A radioactive isotope is released into the air as an
industrial by-product. This isotope is not very stable due to
radioactive decay. Twenty percent of the original radioactive material
loses its radioactivity after each month. If

(a) how much radioactive material is in the atmosphere at the end of the tenth month?

(b) In the long run, i.e., if the situation goes on forever, what will be the amount of this radioactive isotope in the atmosphere at the end of each month ?

**8. **Each day at noon a hospital patient receives * 20
mg* of a
drug. This drug is gradually eliminated from the system in such a way
that over each

(a) How much of the drug is present in the patient's system immediately after noon on the eighth day of treatment ?

(b) If the regime is continued indefinitely, at what level will the amount of the drug in the patient's system stabilize, measured immediately after noon every day ?

**9**.
Use geometric series to find simple fractions that equal the following
repeating decimals:

(a)** A = 0.27777...**

(b) **B = 0.27272727...**

**
10. **At the end of year

(a) Assuming oil consumption increases at this rate in the future, how long will the reserves last ?

(b) Assuming oil consumption increases at
the rate of * 0.8%* per
year in the future, how long will the reserves last ?

^{*}. According to www.bp.com/energy/world_stat_rev/oil/reserves.asp