Preface: Some thought experiments. We
begin this chapter with some model contexts.
[Motion based model] Think of driving a car along a road.
Your odometer
is broken but your speedometer still works. You record only where the
car
is when you begin your trip and you make a table of the car's speed
(velocity)
at regular time intervals during the trip. [Or perhaps you can record
the
velocity at every time from the beginning to the end of the trip].
Based
on this information do you think you could give your location
approximately
(or exactly) at any time during your travels? As a practical matter, if
you have the time to repeat the trip this shouldn't be too difficult.
With
the information you recorded you can reproduce the trip with some
accuracy.
This time use a car with an odometer and let someone else do the
driving
according to the recorded velocities while you record your
position.
For an hour trip this might be more time than you want to spend on this. With a little thought and few arithmetic calculations, you should become up with a reasonable estimate of where you were at the same points in time where you recorded the velocities. 

Graph of Velocities Read from Speedometer during Trip 

[Motion based model] Next suppose you are driving on a highway through a major city. (Los Angeles, Chicago, New Orleans, Miami, or New York will do). You are trying to explain to your passenger why you are travelling at your current speed. What would you include in your explanation? The time of day (6 a.m. or 6 p.m.)? Or the position in the road ( before a major interchange, on a stretch where construction is under way, or on a section where two additional lanes have been added)? Or perhaps the speed at which other vehicles are travelling ( you are travelling up a hill behind two large trucks)? Based on this information can you tell your passenger where you will be in the next few seconds, minutes, hours? How accurate will your predictions be? 

[Production based model.] Think of manufacturing a very desirable product, "good stuff." As your production increased you forgot to keep track of your accumulated costs but you did keep your receipts showing you the marginal rates you were charged for materials and labor. Based on this information, do you think you could give your accumulated costs approximately (or exactly) at any stage of the production? As a practical matter, this is what accountants do in reviewing the accumulation of costs. With the information you have retained you should be able to reproduce the accumulation of costs with great accuracy. 

Graph of Marginal Cost per Unit Produced 
[The Dart Model] Suppose you are throwing a dart at a region
in the
plane and measuring a random variable X determined by the position of
the
dart. X has values only between 0 and 10. Unfortunately you are not
given
the distribution function of X. [Recall that for any real number A
between
0 and 10 the distribution function F, would give you F(A), the
probability
that the value of X is less than or equal to A.] Instead you have been given a graph, an algebraic expression and a table for the probability density function of X. [Recall that for any real number A between 0 and 10 the probability density function of X, f, gives you f(A), the derivative of F at A, which is approximately the ratio of the probability of the random variable X being in an interval [A, A+h] to the length of that interval, h.] Based on this information do you think you could estimate the probability that the value of X being less 5? Recognizing that F(0) = 0 and F(10) = 1, could you estimate F(A) for any A between 0 and 10? As a practical matter, you might break the interval into10 pieces of equal length and then estimate the probability of X being in each of those interval from the density of X at some point in that interval. Then for A, just add up the probability estimates for X being in an interval that has values less than or equal to A. 

Graph of data for f(A) The Probability Density Function for X 
The mathematical investigation of rates using calculus is the defining objective for the study of differential equations. This is one of the most important and, therefore, most studied parts of mathematics. It will be a major theme for the remainder of this book. You should note that this is not our first discussion of rates and related contexts. In our study of implicit differentiation and problems with related rates we encountered many equations involving derivatives of varying complexity. Often the derivative in these studies depended on both the x and y variables.
For example, implicitly differentiating the equation y^{3 }= x^{2} with respect to x leads to the derivative equation, dy/dx = 2x/3y^{2}. Differentiating the same equation with respect to the variable t (which is not expressed in the equation) leads to the derivative equation 3y^{2 }dy/dt = 2x dx/dt. [In fact, we could also express these last two situations with one equation using differentials: 3y^{2} dy = 2x dx.]
In this chapter we begin our examination of differential equations and some of the basic solution concepts for these equations. In doing this we will use our previous work on the derivative (and differentials) to explore these concepts visually, numerically, and symbolically. Our purpose is to develop some systematic methods (a calculus) for solving differential equations. As we proceed through the chapter we will follow a recurrent path of
A differential equation (D.E.) is
an
equation involving derivatives in its statement. Here are some
examples:
(i) dy/dx = x^{ 2}  (ii) y'  x + x^{ 2} = 3xy 
(iii) y' + xy = (y')^{ 2}  (iv) f ''(x) = 5 
(v) p''(x) +xp'(x) + p(x) = 0  (vi) D_{t} f(t) + t f(t) = 3 
(vii) y'' + y' = 5 + y'''  (viii) du/dt + dv/dt = 5 
In some situations a differential equation may be expressed using differentials! For example, the equation dy/dx = x^{2}/y might be expressed as dy = x^{2}/y dx or as y dy = x^{2} dx.
Differential equations for motion were among the early motivations for developing the calculus. Today the need to analyze and control change in complex situations is a major reason why the study of calculus and differential equations is considered so important. These equations are used to study everything:
I am sure you know what it means to solve an equation in algebra. It means to find all those values chosen from a given set of possible solutions that will make the given equation true. So, for example, 2 and 2 are the solutions from the set of integers for the equation x^{2 }= 4. Similarly, the equation sin(t) = 0 has an infinite set of solutions when we allowing real numbers as possible solutions, namely {`t = 2 pi k` where k is any integer}, but this same equation has only one solution that is a rational number, namely 0. 
As a second example you can check that `y = sqrt{25  x^2}` on the interval (5,5) and `y = sqrt{25  x^2}`on the interval (4,4) are both solutions to the equation dy/dx = x/y. With these examples we can identify two key issues in the study of differential equations.
Issue I: When does a differential equation have a solution?We observe as a first response to these issues that it is easier to check whether or not a function is a solution to a given differential equation than it is to find a solution.
Issue II: How unique is the solution of a given differential equation?
The general solution to a differential equation is the family of all functions that are solutions to a given differential equation. For example, the general solution to the differential equation dy/dx = x^{ 2 }is the family of functions of the form y = x^{3 }/ 3 + C where C is a constant. [This statement should seem plausible at this stage. It will be justified in Section IV.B.1.] The general solution to the equation dy/dx = x/y is the family of functions of the form `y = +sqrt{a^2  x^2}` on the interval (a, a) where a>0 is a constant. [This statement will be justified in Chapter VII.]
We can now identify a third key issue in the study of
differential equations,
one that focuses not on existence or uniqueness of a solution, but on
the
ability to describe a solution in familiar terms.
Issue III: Is it
possible to describe the general solution to a differential equation as
some elementary function?
This is not always possible. For example, there is no description
possible
for the differential equation dy/dx = sin(x^{2})
as an elementary function.[This result is not easy to justify.] For
that
reason we are also interested in describing the general solution
graphically
and estimating values for solutions numerically.
This is usually read as "the (indefinite) integral of P(x) with respect to x " or more simply "the (indefinite) integral of P." [The word "indefinite" is frequently omitted.]
P(x) is called the "integrand" of the indefinite
integral.
[The significance and utility of this notation will become more
apparent
later.] The process of finding the general solution in this case is
called
"integration."
Continuing with our example of the differential equation dy/dx = x^{ 2}, we suppose that the general solution is y = x^{ 3}/3 + C. If we use the initial condition y(0)=7 with this situation, it must be that
Looking at the differential equation dy/dx = x/y,
we
suppose
that the general solution is `y = +sqrt{a^2  x^2}`
where
a is a constant. If we use the condition y(3)=4 with this
situation,
it must be that `4 = y(3) = +sqrt{a^2  3^2}` ,
so `16 = a^2  9` and `a^2 = 25` and
therefore
`y = +sqrt{25  x^2}` is
the particular solution to this differential equation with the given
condition.
Notice that in each of these examples there is only one function
that
will satisfy both the differential equation and the given initial
condition,
i.e., the solution in these special situations is unique.
differential  equation  boundary 
solution  antiderivative  particular 
condition  integral  general 
initial  indefinite  unique 
1. For each differential equation determine whether or not the
given
function is a solution.
a. dy/dx = 3x^{ 2}  2x + 5;  y = x^{ 3}  x^{ 2} + 5x  8. 
b. y' + 2y = x^{ 2} + 3x + 5;  y = x^{ 2} + x + 4. 
c. dy = 12x cos(x^{ 2}) dx;  y = 6 sin (x^{ 2}) + 25 
d. y dx + x dy = 4x^{ 3 }dx;.  y = x^{ 3} 
e. y''= 12x^{ 2}  sin(x) + 2;  y = x^{ 4} + sin(x) + x^{ 2} + 3x + 5 
f. P''(t) + P(t) = 0;  P(t) = 5 sin(t)  10 cos(t) 
g. d^{2 }z/du^{2}  9z = 0;  z = e^{3u} 
h. g'(t)= 2/t + 3e^{3t };  g(t) = 2 ln(t)+ e^{3t} 

0  .5  1  1.5  2  2.5  3  3.5  4  4.5  5  5.5  6 

0  30  25  30  10  15  25  10  25  40  50  50  30 
3. In the figure below is the graph of the rate at which water was used during a twenty four hour period in the town of USH. Based on the graph draw a sketch of a graph showing the amount of water used as a function of time during the same period. Estimate the amount of water consumed during the twenty four hour period.
4. Suppose that the general solution to the differential equation dy/dx = 3x^{ 2} + 5 is given by y = x^{ 3} + 5x + C where C is a constant. Determine the value of C that will give a solution to this differential equation with the given condition.
5. Suppose that the general solution to the differential equation y''+ y = 0 is given by y = Asin(x) + Bcos(x) where A and B are constants. Determine the values of A and B that will give a solution to this differential equation with the given conditions.
In problems 6 and 7, do not try to solve the differential equations. Without knowing explicitly what y is as a function of x, but using knowledge of its derivative from the differential equation, there is still much you can say about y in these problems..
6. Assume y is a solution to the differential equation dy/dx=
x/(x^{
2} + 1) with y(0)=2.
(a) Using just the given information, find any local extreme points
for y and discuss the graph of y,
including
the issue of concavity.
(b) Using the differential, estimate y(1) and y(1).
(c) Change the initial condition to y(0) = 3 and then y(0)=
1.
How
will these different conditions affect the graph of y? How
do they affect the estimates for y(1) and y(2) made in
part
b?
7. Assume y is a solution to the differential equation dy/dx=
y
 1 with y(0)=2 and y(x) >1 for all x.
(a) Using just the given information, discuss the graph of y
including the issue of concavity.
(b) Using the differential, estimate y(1/2) and y(1/2).
(c) Suppose that y(x) <1 for all x and y(0)
=
0.
How will these different conditions affect the graph of y?
How do they affect the estimates for y(1/2) and y(1/2)
in
part b?
8. Match each indefinite integral with the family of functions
that
best describes it.
(a) ∫ x^{3} dx  (i) x^{2}/6+C 
(b) ∫ 3x dx  (ii) (3/4)x^{4/3}+C 
(c) ∫ x^{}^{3} dx  (iii) (1/4)x^{4 }+C 
(d) ∫ x^{1/3} dx  (iv) (3/2)x^{2}+C 
(e) ∫ x/3 dx  (v) x^{2}/2+C 
(vi) 3x^{4/3}+C 
10. Match the following statements with the differential equation that best represents the situation described.
(a) As the runner approached the end of the race, she slowed her
rate
steadily so that when she crossed the finish line at 20 seconds she was
travelling at 7 meters per second.
(i) dr/dt = 7t+ 20 (ii) r'(t)=kt + 7(t20)
(iii)
dr/dt = k(t20) + 7
(b) The population of cells had a biomass of 30 milligrams when
the
experiment began. The biomass grew a rate proportional to the mass
present.
(i) P'(t) = k t (ii) P'(t) = k P(t)
(iii)P'(t)=
k (30 + P(t))
(c) The temperature of the coffee was initially 373 degrees
Kelvin while
the room was only about 333 degrees.. The coffee's temperature
decreased
at a rate proportional to the difference between its temperature and
that
of the room.
(i) dT/dt = k (T333) (ii) dT/dt = k ( 373t
 333) (iii) dT/dt = k/(t333)
(d) After 10 minutes the student was learning the material more
slowly
as time went on, so that his rate of learning was then decreasing
inversely
with the square of the time he had spent reviewing.
(i) dL/dt = k (t +10)^{2} (ii) DL(t) =
k/t^{2 }(iii) L'(t) = k/(t+10)^{2}
(e) The profits from the investment were increasing more slowly
as time
went on so that the rate appeared to decrease inversely proportionately
with the profits at any given time.
(i) P'(t)=k  P(t) (ii) DP(t) = k/t (iii)
dP/dt = k/P
(f) The marginal revenues from the production of more than 100
faucets
decrease proportionately with the number of additional faucets
produced.
(i) dR/dx = kx (ii) DR(x) = k(x+100)
(iii) R' = k(x100)
11. Which of the following equations mean the same thing? Explain your response.
12. Use the interpretation of the derivative as the velocity of a moving object to explain the following statement: If two functions have the same derivative at every point then the difference between the values of the functions at every point will be the same number.
13. Describe three contexts where the rate at which something is used is monitored to predict how much will be used in the future. Discuss how do these predictions affect planning?
14. The following table gives the speed of cars on a rural
highway at
the given time of day and the mile maker.
\ time mile marker 
5 a.m.  8 a.m.  11 a.m. 
10 mi.  55 mph  40 mph  50 mph 
20 mi.  30 mph  20 mph  30 mph 
30 mi.  40 mph  25 mph  30 mph 
15. Sometimes it is possible to show that a differential equation cannot be solved by functions of designated type. For each of the following differential equations do not try to find a solution. Based on the equation explain why there can not be any polynomial function that would satisfy the equation.
a) f '(x) = 5 f(x) and f(0)
=
2.
[Hint: Find a formula for f ^{(n)}(x). Show
that
f
^{(n)
}(0)
is not 0 for any n^{th }order derivative of f.]
b) f ''(x) = 5 f(x) and f(0) =
2.
c) dy/dx = 1/(1 + x^{2}) and y(0)
= 2.
16. Suppose X is a random variable with range [0,10]. Give a differential equation that shows the relation of the probability density function f (x) and the distribution function F(x) for X. Explain briefly why the boundary conditions for this differential equation are F(0) = 0 and F(10)=1. Generalize this situation to a random variable with range [A,B].