The Language of Differential Equations

We are all used to equations like

2x + 3 = 17

x² = 4

that tell us something about numbers. For example, the second equation,

x² = 4,

tells us that x is a number whose square is 4. Differential equations tell us something about functions. The following equations are all differential equations.

p' = 0.03 p
y'' = -32
T' = k(A - T)

The first equation might describe a population that is growing at the rate of 3% per year. The second equation might describe the height of a falling object that is accelerating at the rate of -32 feet per second per second. The third equation, Newton's Model of Cooling, might describe the temperature of an object in a room whose temperature is A.

The first and third equations are first order differential equations because they involve the first derivative of a function but no higher derivatives. The second equation is a second order differential equation because it involves the second derivative but no higher derivatives.

The language of differential equations allows us to express ideas involving change and to work with those ideas -- testing them, using them to make predictions, and communicating them. The first five modules in this chapter on differential equations provide an introduction and overview of the subject.

Using the language of differential equations (this module).
Looking at differential equations visually
Estimating the solution of an initial value problem visually
Estimating the solution of an initial value problem numerically
Guess-and-Check -- Finding exact solutions of differential equations

English (or French, Latin, Spanish, or any other language) and differential equations are both languages that we use to talk about the real world -- to express, communicate, and manipulate ideas. Just as French and English both have words -- red and rouge -- for the real world color shown below

English and differential equations both have "words" -- velocity and f'(t) -- for expressing the idea shown in the movie below.

In ordinary English we use the word red for any of the colors shown in the bar below but a house painter or an artist must use words more precisely. Similarly, when we are speaking using the language of mathematics we often can be and want to be more precise than in ordinary everyday English.

Words like speed and velocity are often used interchangably in ordinary English but we need to be more careful. Here is a glossary of some of the "words" we use in the language of differential equations and in careful English.

location or
position We often use a function, f(t), to represent the location or position of a moving object at time t. For example, if we drop an object a time 0 from a height of six feet then f(t) might represent its height measured in feet at time t measured in seconds.

velocity The velocity of a moving object is represented by the derivative, f'(t), of its position. If we drop an object from a height of six feet at time t = 0 then f(0) = 6 because its height at time 0 is six feet and f'(0) = 0 because at the instant we let it go its velocity is zero.

speed The word speed is used for the magnitude of the velocity -- that is, |f'(t)|.

acceleration The acceleration of an object is the rate at which its velocity is changing -- that is, f''(t).

In general, of course, the derivative of a function represents the rate at which the function is changing or the sensitivity of one variable to changes in another variable. The words we use in English sometimes depend on the context. For example, in economics people often talk about the marginal price of a product. The graph below shows a typical relationship between the quantity of a product and its price.

The graph above shows a typical phenomenon -- for small quantities the marginal price or unit price is larger than for large quantities. We see this visually -- the red part of the graph is steeper than the blue part. This often happens because there are "economies of scale."

In many situations, we talk about the percentage rate of change or the relative rate of change. For example, we often say that population is rising at the rate of 2% per year or that the rate of inflation (the rate at which prices are rising) is 3% per year. If f(t) is a function then its relative rate of change is

For example, if population is rising at the rate of 2% per year we can write


                              p'(t)
                              -----  = 0.03
                               p(t)


                              p'(t) = 0.03 p(t)

In general, a differential equation describes how a function is changing -- for example, the differential equation above says that the population is growing at the rate of 3% per year. We usually need additional information -- for example, the population at a given time. For a first order differential equation we need one additional piece of information or initial condition. For a second order differential equation we need two additional pieces of information or initial conditions. A complete description of a situation including a first order differential equation and one initial condition or a second order differential equation and two initial conditions is called an initial value problem. For example, the initial value problem

f'(t) = 60, f(0) = 40

describes the travels of a car that passes mile marker 40 on a superhighway at time t = 0 and then drives steadily at 60 miles per hour.

The initial value problem

h''(t) -32, h(0) = 6, h'(0) = 0

describes the travels of a baseball that is dropped at time t = 0 from a height of six feet. Notice h'(0) = 0 because at the instant the baseball is realised it is still.

Translate each of the following English descriptions into the language of differential equations and then draw a graph describing the same situation. You will not need to use your CAS window for these problems.

A car is driving at a steady velocity of 60 miles per hour northward along an interstate highway. At time t = 0 it is located 23 miles north of the start of the highway. answer
A mail order company charges a flat rate of $12.00 for shipping, $10.00 per pound for the first five pounds of coffee and $8.00 per pound for additional coffee. answer
One proponent of the flat tax has proposed that single taxpayers pay no income tax if their income is below $8,000 and then 17% of any income above $8,000. answer
The population of a city is increasing at the rate of 1,200 people per year due to immigration. Its current population is 50,000. answer

In the problems above, the graphs were easy to draw because they were made up of one or more straight line segments. The next problems we look at are more interesting and more complicated. Your CAS window shows you how to draw graphs of functions described by differential equations using a numerical method similar to Euler's Method, which we describe later both graphically and numerically. For now we let your computer algebra system do the work.

Example:

Newton's Model of Cooling says that the rate at which an object cools is proportional to the difference between its temperature and the ambient temperature (the temperature of its immediate environment). If we denote the temperature of the object at time t by T(t) this leads to the differential equation

 
                              dT
                              -- = k(A - T)
                              dt

where A is the ambient temperature. The constant k depends on the object. If the object is small and poorly-insulated then k will be large but if the object is large and well-insulated the value of k will be small.

Suppose that a cup of coffee starts out at time t = 0 at a temperature of 80 degrees Celsius in a room whose temperature is 23 degrees Celsius. Suppose that k = 0.05. This situation can be described by the initial value problem.

 
                         dT
                         -- = 0.05 (23 - T),  T(0) = 80.
                         dt

The function described by this initial value problem is shown in the graph below.

Example:

Suppose a baseball is thrown straight upward from a height of five feet with an initial velocity of 30 feet per second. This situation is described by the initial value problem


                         h''(t) = -32,  h(0) = 5,  h'(0) = 30

The function described by this initial value problem is shown in the graph below.

Use your CAS window to draw the graphs requested in the problems below.

Suppose that at time t = 0 the furnace in a building turns on so that the ambient temperature in one room is given by the function
```
 
A(t) = 65 + 0.10 t,  0 <= t <= 50. 
```
Suppose that a cup of coffee in the room starts out at time t = 0 at 170 degrees and that its temperature changes according to Newton's Model of Cooling
```
T' = 0.15 (A(t) - T)
```
Draw a graph showing the temperature of the cup of coffee for 0 <= t <= 50.
Suppose that the temperature in a room is fluctuating because the furnace keeps cycling on-and-off so that the ambient temperature in one room is described by the function
```
A(t) = 67 + 4 sin (t/5)
```
Suppose that a cup of coffee in the room starts out at time t = 0 at 170 degrees and that its temperature changes according to Newton's Model of Cooling
```
T' = 0.15 (A(t) - T)
```
Draw a graph showing the temperature of the cup of coffee.
When an object falls it is subjected to two forces -- the force of gravity and a force produced by friction. If the object is not too blunt the force due to friction is proportional to velocity. This leads to the differential equation
```
h''(t) = -32 - p h'(t)
```
where p is a positive constant depending on the size and shape of the object.
Suppose that p = 0.5 and draw a graph showing what happens when this object is dropped at time t = 0 from a height of 1,000 feet.
When an object falls it is subjected to two forces -- the force of gravity and a force produced by friction. If the object is very blunt the force due to friction is proportional to the square of its velocity. This leads to the differential equation
```
h''(t) = -32 + p h'(t)^2
```
where p is a positive constant depending on the size and shape of the object. This differential equation applies when the object is falling. Why? What change would be needed if the object were rising?
Suppose that p = 0.5 and draw a graph showing what happens when this object is dropped at time t = 0 from a height of 1,000 feet.

location or position	We often use a function, f(t), to represent the location or position of a moving object at time t. For example, if we drop an object a time 0 from a height of six feet then f(t) might represent its height measured in feet at time t measured in seconds.
velocity	The velocity of a moving object is represented by the derivative, f'(t), of its position. If we drop an object from a height of six feet at time t = 0 then f(0) = 6 because its height at time 0 is six feet and f'(0) = 0 because at the instant we let it go its velocity is zero.
speed	The word speed is used for the magnitude of the velocity -- that is, \|f'(t)\|.
acceleration	The acceleration of an object is the rate at which its velocity is changing -- that is, f''(t).