WorkSheet 2
Review of the Differential Equations Material
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1. Methods of solving differential equations.
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Separation of Variables
Integrating factor
Know solution to dy/dx = ay and dy/dx = ay, y(0) = c
Know solution to dy/dx = ay + b and dy/dx = ay + b, y(0) = c
Know Maple syntax for solving differential equations.
2. Exponential Growth/Decay problems
— Set up for continuous compounding problems (i.e. start from A(t+∆t)−A(t)
and end with the differential equation dA/dt = rA).
— Know how to use the exponential equation to solve continuous compounding,
bacteria growth, and radio active decay problem
3. Newton’s Law of cooling
— Know the statement leading to the differential equation.
— Know how to use separation of variables (or simply know the solution since
it is of the form dx/dt = ax + b) to solve dT /dt = a(T (t) − Ts )
4. Salt tank problems (one tank)
— Rate in = rate out (Easy to solve)
— Rate in 6= rate out (Can not use separation of variables - will need the integrating factor technique)
5. Falling Body problems.
— Assume that gravity is the only force (easy integration problem)
— Assume that wind resistance is also a force on the object (equation is now a
differential equation).
— Know how to set up these problems in different coordinate systems
6. Interest rate problems
— Continuous compounding
— Depositing or withdrawing continuously
7. Direction Fields
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Simple ones by hand
Using Maple
Equilibrium solutions
Stability of equilibrium solutions
8. Population problems
— Use the direction field to analyze
— Solve with Maple
9. Numerical Solutions
— Know how to derive Euler’s method
— Know how to use Euler’s method to generate a couple values of the function.
— Understand the maple document which gives other numerical methods
Problems related to First Order DEs
1) What is the general solution to
dy
= 3y?
dx
dx
= −2x, x(0) = 15?
dt
dx
3) What is the solution to
= αx, x(0) = 5, x(2) = 1?
dt
2) What is the solution to
dy
= 3y.
dx
dy
√
5) Use the separation of variable technique to find the general solution to
= 3x y.
dx
dx
6) Use the separation of variable technique to find the solution to
= 3t2 x2 , x(1) = 1.
dt
7) 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 bacteria and two hours later there are 2.1 grams,
how much will you have after 5 hours?
4) Use the separation of variable technique to find the general solution to
8) 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?
9) 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.
10) Suppose that the situation in (9) 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
difference between the temperature of the object and the temperature of the surrounding
medium. What DE describes the temperature on this new planet?
11) 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, find the amount of salt in the tank at any future
time t.
12) 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 bacteria and three hours later
there are 3.5 grams, how much will you have after 7 hours?
13) 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?
14) If a 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.
15) 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 10 gallons per minute and the
mixture leaves the tank at the same rate, how long will it take to flush 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?
16) 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
P on the ball and also that the mass of the ball is 2 kg. First
use Newton’s Law ( ma = f orces) to set up the problem as a differential equation. Then
solve the problem (find 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.
17) 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
P a speed of 70 ft/sec. Suppose the ball weighs
2 lbs. First use Newton’s Law ( ma =
f orces) to set up the problem as a differential
equation. Then solve the problem (find 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?
18) 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, find 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.
19) (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?
20) 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?
21) 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?
22) 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?
23) 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?
dy
24) Solve
= 2y + 4, y(0) = 3 in three different ways: separation of variables, integrating
dx
factor method, and using the structure of this non-homogeneous problem.
25) Even the simple exponential growth DEs can be solved using the integrating factor
dx
= 3x, x(0) = 8 using the integrating factor method.
method. Solve
dt
dy
26) Solve
= 2xy − 3x
dx
dy
27) Solve
= −y + x2 , y(0) = 2
dx
√
dx
x
28) Solve
= 2√ + t
dt
t
dx
29) Solve
= −2x + sin(t)
dt
dy
y
30) Solve
+3 =4
dx
x
31) 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 overflows? Use whatever method is appropriate for solving
this problem. Also give a graph of the solution.
32) 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 overflows? Use
whatever method is appropriate for solving this problem. Also give a graph of the solution.
33) 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. Use a position
coordinate system where up is positive and with the origin at the middle of the building.
a) Assuming that only gravity acts on the ball, find 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.
c) Change the resistive force in (b) so that the resistance force is 0.8 times the square of the
speed of the ball. Now determine how long the ball is in the air. Be careful. I think this
problem requires a careful setup.
34) (a) 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 6 gallons per minute, how much salt is in the tank at time t?
Sketch the solution.
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?
Sketch the solution.
35) (a) 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?
(b) 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?
36) The radioactive isotope plutonium 241 decays so as to satisfy the differential equation
dx
= −0.0525x where x is measured in milligrams and t in years. Determine the half—life of
dt
plutonium 241.
37) 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?
38) 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 rate of
4 gallons per minute, how much salt is in the tank at the time the tank overflows? Use
whatever method is appropriate for solving this problem. Also give a graph of the solution.
39) Given the IVP dx/dt = f (t, x), x(0) = x0 , show how Euler’s method gives an approximation x1 to x(h) and x2 = x(2h). Be sure to use a picture and clearly show the algorithm
to find x1 and x2 .
40) Find x1 and x2 using Euler’s method given that dx/dt = 2tx − ex cos(t), x(0) = 1 using
h = 0.05.
41) Find x1 and x2 using Euler’s method given that dx/dt = t2 +x2 , x(0) = 1 using h = 0.05.
42) An animal sanctuary had an initial population of 50 animals. After 2 years the population
was 62, while after 4 years it was 76. Using the logistic population model, determine the
equilibrium solution and the number of animals in the sanctuary after 20 years.