Calculus II, MA 112
WorkSheet 8
dy
1) What is the general solution to
= 3y?
dx
2) What is the solution to
dx
= ¡2x; x(0) = 15?
dt
3) What is the solution to
dx
= ®x; x(0) = 5; x(2) = 1?
dt
January 25, 1999
4) Use the separation of variable technique to derive the general solution to
dy
= 3y:
dx
5) Set up, but don’t solve during class, the di¤erential equation (along with any additional
infomation that you need) which describes each of the following. Be sure to describe your
function in English (eg. A(t) is the amount of money in the account at time t):
a) A certain type of bacteria is known to grow at a rate proportional to the amount that is
present. If you start with 2 grams of the bacterial and two hours later there are 2.1 grams,
how much will will you have after 5 hours?
b) Suppose that 15 grams of Einsteinium were released (the half life of Einsteinium is 270
days). How long will it take for the 15 grams to decay to 10 grams?
c) If glass of cold water (40 degrees) is placed in a room whose temperature is 70 degrees,
how long will it take for the glass of water to heat up to 55 degrees? Use Newton’s Law of
Cooling and assume that the temperature after 10 minutes is 44 degrees.
d) Suppose that the situtation in (c) is taking place on some other planet where the Law of
Cooling state that the rate of change of the temperature is proportional to the square of the
di¤erence between the temperature of the object and the temperature of the surrounding
medium. What DE describes the temperature on this new planet?
e) A tank 100 gallon tank contains a solution of water and salt with the initial amount of
salt being 25 lbs. If pure water enters the tank at a rate of 5 gallons per minute and the
mixture leaves the tank at the same rate, …nd the amount of salt in the tank at any future
time t:
Calculus II, MA 112
WorkSheet 9
dx
1) What is the general solution to
= ¡2x?
dt
2) What is the solution to
dy
= 5y; y(0) = 4?
dt
3) What is the solution to
dx
= ®x; x(0) = 5; x(2) = 1?
dt
January 26, 1999
4) Use the separation of variable technique to derive the general solution to
dy
= ®y:
dx
5) After setting up the following, work on solutions to yesterday’s #5 and today’s #5.
a) A certain type of bacteria is known to grow at a rate proportional to the square root of
the amount that is present. If you start with 3 grams of the bacterial and three hours later
there are 3.5 grams, how much will will you have after 7 hours?
b) Suppose that 10 grams of radium were released (the half life of Radium is 1620 years).
How long will it take for the 10 grams to decay to 5 grams? to 2 grams?
c) If cup of hot chocolate (180 degrees) is placed in a room whose temperature is 70 degrees,
how long will it take for the cup to cool to 100 degrees? Use Newton’s Law of Cooling and
assume that the temperature after 5 minutes is 150 degrees.
d) A tank 500 gallon tank contains a solution of water and salt with the initial amount of
salt being 100 lbs. If pure water enters the tank at a rate of 10 gallons per minute and the
mixture leaves the tank at the same rate, how long will it take to ‡ush the tank (assume that
this means that 98% of the original salt is gone). What would the answer be if we started
with S lbs of salt?
Calculus II, MA 112
WorkSheet 10
January 28, 1999
This is a gentle introduction to falling body problems. We will assume that the gravitational
constant g (9.8 m/sec2 or 32.2 ft/sec2 ) is always positive. When working a falling body
problem, I will always want to see the coordinate axis that you are using.
1) Suppose that you are on top of a 60 m building and you throw a ball upwards (from the
edge so that it returns to the ground) with a speed of 50 m/sec. Now we will assume that
gravity is the only force acting on the ball and also that the mass of the ball is 2 kg. First
P
use Newton’s Law ( ma = f orces) to set up the problem as a di¤erential equation. Then
solve the problem (…nd the position x as a function t) and determine how high the ball goes
and how long it is in the air.
a) Use a position coordinate system where up is positive and with origin at the top of the
building.
b) Use a position coordinate system where up is positive and with origin on the ground.
c) Use a position coordinate system where down is positive and with origin at the top of the
building.
2) Suppose that you are on top of a 100 ft building and you throw a ball downwards (from the
edge so that it goes to the ground) with a speed of 70 ft/sec. Suppose the ball weighs 2 lbs.
P
First use Newton’s Law ( ma = f orces) to set up the problem as a di¤erential equation.
Then solve the problem (…nd the position x as a function t) and determine how long it takes
the ball to reach the ground. What is the ball’s velocity when it hits the ground?
a) Use a position coordinate system where up is positive and with origin at the top of the
building.
b) Use a position coordinate system where up is positive and with origin on the ground.
c) Use a position coordinate system where down is positive and with origin at the top of the
building.
Calculus II, MA 112
WorkSheet 11
February 2, 1999
1) Suppose that you are on top of a 20 m building and you toss a 2 kg ball upwards (from
the edge so that it returns to the ground) with a speed of 25 m/sec. Us a position coordinate
system where up is positive and with origin on the ground.
a) Assuming that only gravity acts on the ball, …nd how long the ball is in the air
b) In addition to gravity, assume that there is a resistance force whose magnitude is 1.2
times the speed of the ball. Now determine how long the ball is in the air.
2) (a) A 500 gallon tank contains a solution of water and salt with the initial amount of
salt being 100 lbs. If pure water enters the tank at a rate of 8 gallons per minute and the
mixture leaves the tank at the same rate, how much salt is in the tank at time t?
b) Same setup as (a) except there is salt in the water that enters the tank. Assuming that
there is 1/2 lb of salt per gallon entering the tank, how much salt is in the tank at time t?
Calculus II, MA 112
WorkSheet 12
February 4, 1999
1) A 250 gallon tank contains a solution of water and salt with the initial amount of salt
being 60 lbs. If pure water enters the tank at a rate of 5 gallons per minute and the mixture
leaves the tank at the same rate, how much salt is in the tank at time t?
2) A 250 gallon tank contains a solution of water and salt with the initial amount of salt
being 60 lbs. If pure water enters the tank at a rate of 5 gallons per minute and the mixture
leaves the tank at the rate of 7 gallons per minute, how much salt is in the tank at time t?
3) A 250 gallon tank contains a solution of water and salt with the initial amount of salt
being 60 lbs. If a brine solution containing 1/2 lb per gallon enters the tank at a rate of 5
gallons per minute and the mixture leaves the tank at the same rate, how much salt is in
the tank at time t?
4) A 250 gallon tank contains a solution of water and salt with the initial amount of salt
being 60 lbs. If a brine solution containing 1/2 lb per gallon enters the tank at a rate of 5
gallons per minute and the mixture leaves the tank at the rater of 7 gallons per minute, how
much salt is in the tank at time t?
Calculus II, MA 112
WorkSheet 13
February 5, 1999
The Integrating Factor Technique
Step 1: Put the di¤erential equation in the form
dy
dx
+ f (x)y = g(x) (or
+ f (t)x = g(t))
dx
dt
dy
dy
= 3xy ¡ 4 as
¡ 3xy = ¡4 (so here f (x) = ¡3x and g(x) = ¡4)
dx
dx
dx
dx 2
Example: Write
= ¡t2 x + sin(t) as
+ t x = sin(t) (so here f (t) = t2 and g(t) = sin(t))
dt
dt
Step 2: Find the antiderivative of f (x): You do not need the \ + C”:
Example: Write
Step 3: If F (x) is the antiderivative of f(x); multiply both sides of the DE (in the proper
form) by eF (x) : (this function is called the integrating factor).
eF (x) (
dy
dy
+ f(x)) = g(x)eF (x) or eF (x)
+ f (x)eF (x) = g(x)eF (x)
dx
dx
³
Step 4: Notice that the left hand side is the derivative of a product. yeF (x)
´0
= g(x)eF (x)
Step
integrate
both sides of the this new equation with respect to x:
Z
Z ³ 5: Now
´0
F (x)
ye
dx = g(x)eF (x) dx and notice that the left hand side is simply yeF (x)
Z
Final Step: Now solve for y: From Step 5, we have yeF (x) = g(x)eF (x) dx: After …nding
the antiderivative of the right hand side (remember to attach a “+C”), divide both sides by
eF (x) :
dy
Let’s go through the above steps with the easy example:
= ¡3y + 4
dx
Note that you could have used separation of variable in the previous example. Here’s an
dx
x
example where separation of variables won’t work:
=¡ +4
dt
t
HOMEWORK due Monday, February 8
dy
= 2y + 4; y(0) = 3 in two di¤erent ways: separation of variables and integrating
1) Solve
dx
factor method.
2) Even the simple exponential growth DEs can be solved using the integrating factor method.
dx
Solve
= 3x; x(0) = 8 using the integrating factor method.
dt
3) Solve
dy
= 2xy ¡ 3x
dx
4) Solve
dy
= ¡y + x2 ; y(0) = 2
dx
5) Solve
p
dx
x
= 2p + t
dt
t
6) Solve
dx
= ¡2x + sin(t)
dt
7) A 200 gallon tank initially contains 100 gallons of a solution of water and salt with the
initial amount of salt being 25 lbs. If pure water enters the tank at a rate of 6 gallons per
minute and the mixture leaves the tank at the rate of 4 gallons per minute, how much salt is
in the tank at the time the tank over‡ows? Use whatever method is appropriate for solving
this problem. Also give a graph of the solution.
8) A 200 gallon tank initially contains 100 gallons of a solution of water and salt with the
initial amount of salt being 25 lbs. If a brine solution containing 1/4 lb per gallon enters
the tank at a rate of 7 gallons per minute and the mixture leaves the tank at the rater
of 4 gallons per minute, how much salt is in the tank at the time the tank over‡ows? Use
whatever method is appropriate for solving this problem. Also give a graph of the solution.
Calculus II, MA 112
WorkSheet 14
February 8, 1999
Solve the following di¤erential equations (use whatever technique is appropriate).
dy
dx
1)
= 2y ; y(0) = 3
(2)
= ¡2x ; x(0) = 12
dx
dt
3)
dx
= ¡3x + 4
dt
4)
dx
= x+t
dt
5)
dy
y
+3 =4
dx
x
6) Suppose that you are on top of a 30 m building and you toss a 1.5 kg ball upwards (from
the edge so that it returns to the ground) with a speed of 20 m/sec. Us a position coordinate
system where up is positive and with origin at the middle of the building.
a) Assuming that only gravity acts on the ball, …nd how long the ball is in the air.
b) In addition to gravity, assume that there is a resistance force whose magnitude is 0.8
times the speed of the ball. Now determine how long the ball is in the air.Setup during class
and complete details later for homework.
7) A 200 gallon tank contains a solution of water and salt with the initial amount of salt
being 50 lbs. If pure water enters the tank at a rate of 4 gallons per minute and the mixture
leaves the tank at the same rate, how much salt is in the tank at time t?
b) Same setup as (a) except there is salt in the water that enters the tank. Assuming that
there is 1/2 lb of salt per gallon entering the tank, how much salt is in the tank at time t?
Setup during class and complete details later for homework.
8) Suppose that $1000 is put into a savings account at 8% interest compounded continuously.
How much money is in the account at the end of 1 year, 1.5 years, and 5 years?
9) In addition to putting $1000 into a savings account at 8% interest compounded continuously, you decide to put another $1000 into the account throughout the year. Assume that
this money is being put into the account continuously (for instance, you might be putting
in $1000/52 per week – this would almost be continuous). Now how much money is in the
account at the end of 1 year, 1.5 years, and 5 years?
HOMEWORK due Tuesday, February 9
1) Complete the details of 6b on worksheet 14.
2) Complete the details of 7b on worksheet 14.
3) Complete the details of 7b on worksheet 14.
4) The radioactive isotope plutonium 241 decays so as to satify the di¤erential equation
dx
= ¡0:0525x where x is measured in milligrams and t in years.Determine the half–life of
dt
plutonium 241.
5) Suppose that a person who is just about to retire has saved A0 dollars while working the
past few years. In order to invest this money at a 6% annual interest rate (compounded
continuously) and be able to withdraw $20; 000 per year (assume that withdrawals are made
continuously) for the next 20 years, what is the minimum value that A0 can be?
6) A 300 gallon tank initially contains 200 gallons of a solution of water and salt with the
initial amount of salt being 50 lbs. If a brine solution containing 1/2 lb per gallon enters
the tank at a rate of 6 gallons per minute and the mixture leaves the tank at the rater
of 4 gallons per minute, how much salt is in the tank at the time the tank over‡ows? Use
whatever method is appropriate for solving this problem. Also give a graph of the solution.