MATH 285: Worksheet 1
August 28, 2014
1. Using actual words, answer the following questions.
(a) What is an ordinary differential equation (ODE)?
(b) What is the “answer” when solving a differential equation?
(c) How can you check your “answer”? This means you should never get a
problem wrong! How exciting!
(d) What is the difference between an ODE and an initial-value problem (IVP)?
(e) What is the “answer” when solving an IVP?
(f) How can you check your “answer”?
2. Substitute y = erx into the given differential equations to determine all values of the
constant r for which y = erx is a solution of the equation.
(a) 4y 00 = y
(b) 3y 00 + 3y 0 − 4y = 0
3. Verify that the following functions satisfy the given differential equations and find the
constant C so that y(x) satisfies the initial condition.
(a) y 0 = 2y; y(x) = Ce2x , y(0) = 3
(b) y 0 = 3x2 (y 2 + 1); y(x) = tan(x3 + C), y(0) = 1
4. Differential equations are supremely important to mathematical models. Again using
actual words, answer the following questions.
represents when P
(a) What does the derivative represent? Think about what dP
dt
dy
dT
represents population, dx when y(x) is a function, dt when T is temperature.
(b) What real-world applications can you think of that could be represented as a
differential equation? What is the key relationship you need for a phenomenon
to be able to be represented as a differential equation. (Think part a).
(c) Put the steps of the modeling process in order. (see other handout)
(d) Write a differential equation that is a mathematical model for the following situation:
In a city with a fixed population of P persons, the time rate of change of the number N of those persons infected with a certain contagious disease is proportional
to the product of the number who have the disease and the number who do not.
1
5. Slope Fields
dy
(a) Consider the differential equation dx
= x/2. What is the slope of a solution
y = y(x) when x is 0? 1? -1? 2? -2?
(b) Draw a set of (x, y)-axes, with x and y between −2 and 2. Draw a short line
segment at each integer point that reflects the slope of a solution passing through
that point. This is called a slope field.
(c) Place you pencil at the point (−2, 1) and trace what a solution to the differential
equation through that point should look like.
(d) Repeat the process for the differential equation
dy
dx
= y/2.
(e) What is a slope field good for? Why are we talking about them? Are they
helpful if you know the solution to a differential equation? Can you always find
2
dy
dy
= ex or dx
= sin(x)
). We call slope fields a
the solution? (Hint consider dx
x
qualitative method for analyzing a differential equation.
6. Mythbusters: It is said that if you throw a penny off of the Empire State Building and
it hits someone in the head it will kill them. Let’s see if it’s true.
(a) Model
i. State the acceleration due to gravity in f t/s2 .
ii. The resistance due to air on a falling object is proportional to the objects
velocity. The coefficient of drag on a penny is .33s−1 . State the acceleration
due to drag on the penny.
of the penny in feet per second.
iii. Develop a model for the acceleration dv
dt
(b) Graph, “solve,” and interpret
i. Sketch a slope field for the given differential equation. In particular, pay
= 0.
attention to the points where dv
dt
ii. Sketch some possible solution curves for the differential equation. About how
fast will the penny be going when it hits the ground? Will it actually kill
someone?
7. Match the slope field cards. Each group of cards should have one differential equation,
one slope field, and one description.
8. Discuss strategies for success in this class with your group.
An Ending Thought: Believe you can and you’re halfway there.
– Theodore Roosevelt
2