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 (4 a.m. or 4 p.m.)? or the position in the road ( before a major interchange, on a stretch where construction is going on, 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?
Here's another context. You are on a treasure hunt. Besides telling you where to start and which way to go, the directions tell you the speed you should travel based on the time elapsed since you start your journey and your current position. Would you be able to tell where you were at any time during this trip?
Though these trip illustrations may seem a bit artificial, the situations reflect many real circumstances. Often we know something about the rate at which some measured quantity (e.g., rain, snow, population of owls or fruit flies, people infected with a virus, chemical, or money) is changing. Using these rates we would like to know the quantity present at certain times or we would like to predict when the quantity will reach certain levels (either desirable or undesirable). Here are two key questions: How do the rates control the actual amounts of those quantities as they change and what controls these rates? Scientific investigators and engineers look for relations that determine rates in real contexts along with specific measurements from which the information can be extended, by interpolation or extrapolation.
The mathematical investigation of rates using calculus is part of
the study of differential equations. This is one of the most important
and most studied parts of mathematics. It will be a major theme for the
remainder of this book. This is not our first discussion of rates and related
situations. Implicit differentiation and problems with related rates dealt
with equations involving derivatives of varying complexity. Sometimes the
derivative 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 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.
Interpretation: We have seen that a derivative can be interpreted dynamically as a rate of change (of various orders) or geometrically as the slope of a tangent line. Following the dynamic view, a differential equation can be interpreted as a relation between measured quantities and their rates of change. Understanding these relations is at the heart of most questions in the sciences. A scientist studies not only what things are but also how things change. An understanding of change allows scientists (and engineers) to predict and thereby control different aspects of our world. This is easy to believe in the physical, chemical, and biological science, but it is increasingly true as well in such disciplines as economics, management, geography, and other social sciences. You can find contexts for using differential equations almost anywhere you look.
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: from the forces in the smallest subatomic particles, to the
motion of space craft, from the growth of small cell and insect populations
to the complex interrelations of environmental systems, from the effect
of a sudden cooling on a hot summer day to the global effects of a hole
in the ozone layer of the atmosphere, and from the profitability of a single
business decision to the effect of major fiscal and monetary policy changes
on national and international economics. Be it heat, sound, light, or the
price of butter in Burbank, whatever the subject, there is usually one
or more differential equations useful in its study. It is should be no
surprise then that many disciplines require their students study calculus.
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 = 2pk 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 on the interval (5,5) and on the interval (4,4) are both solutions to the equation dy/dx = x/y. With these examples we can identify one of the two key issues in the study of differential equations.
Issue I: Does a given 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 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.Conventions in Language and Notation:
When the differential equation has the form y' = P(x) where P is a function of x only, the general solution is also called the indefinite integral of P(x) or the antiderivative of P(x) and is denoted
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."
As the remarks in the preface to this chapter indicated, knowing where you start on a trip determines in part where you will be subsequently. Often a differential equation has one or more additional conditions which a solution must satisfy as well, such as y(0) = 7 or y(3) = 4. Additional conditions often arise in applications from information given about the beginning of a situation ("initial conditions") or conditions that must be satisfied at the beginning and end of the process ("boundary conditions"). Such conditions restrict the selection of solutions to only a few particular members from the family of all solutions. A particular (or special) solution to a differential equation is a solution that satisfies any such additional conditions.
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 where a is a constant. If we use the condition y(3)=4 with this situation, it must be that , so and and therefore 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. g'(t)= 2/t + 3e^{3t };  g(t) = 2 ln(t)+ e^{3t} 
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. dy = 12x cos(x^{ 2}) dx;  y = 6 sin (x^{ 2}) + 25 

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

0  40  25  30  10  30  25  40  55  50  50  50  20 
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)?
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)?
8. Match each indefinite integral with the family of functions that
best describes it.
(a)  (i)x^{2}/6+C 
(b)  (ii)(3/4)x^{4/3}+C 
(c)  (iii)x^{4}/4+C 
(d)  (iv)(3/2)x^{2}+C 
(e)  (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 proportionate 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 situations 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 city expressway
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 [0,2]. 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(2)=1. Generalize this situation to a random variable with range [A,B].