7.1. AN INTRODUCTION TO DIFFERENTIAL EQUATIONS
389
Because differential equations describe the derivative of a function, they give us
information about how that function changes. Our goal will be to take this information and
use it to predict the value of the function in the future; in this way, differential equations
provide us with something like a crystal ball.
Differential equations arise frequently in our every day world. For instance, you may
hear a bank advertising:
Your money will grow at a 3% annual interest rate with us.
This innocuous statement is really a differential equation. Let’s translate: A(t) will be
amount of money you have in your account at time t. On one hand, the rate at which
your money grows is the derivative dA/dt. On the other hand, we are told that this rate is
0.03A. This leads to the differential equation
dA
dt
= 0.03A.
This differential equation has a slightly different feel than the previous equation
ds
dt = 4t + 1. In the earlier example, the rate of change depends only on the independent
variable t, and we may find s(t) by integrating the velocity 4t + 1. In the banking example,
however, the rate of change depends on the dependent variable A, so we’ll need some new
techniques in order to find A(t).
Activity 7.1.
Express the following statements as differential equations. In each case, you will need
to introduce notation to describe the important quantities in the statement so be sure
to clearly state what your notation means.
(a) The population of a town grows continuously at an annual rate of 1.25%.
(b) A radioactive sample loses 5.6% of its mass every day.
(c) You have a bank account that continuously earns 4% interest every year. At the
same time, you withdraw money continually from the account at the rate of
$1000 per year.
(d) A cup of hot chocolate is sitting in a 70 ◦ room. The temperature of the
hot chocolate cools continuously by 10% of the difference between the hot
chocolate’s temperature and the room temperature every minute.
(e) A can of cold soda is sitting in a 70 ◦ room. The temperature of the soda warms
continuously at the rate of 10% of the difference between the soda’s temperature
and the room’s temperature every minute.
⊳
389
Because differential equations describe the derivative of a function, they give us
information about how that function changes. Our goal will be to take this information and
use it to predict the value of the function in the future; in this way, differential equations
provide us with something like a crystal ball.
Differential equations arise frequently in our every day world. For instance, you may
hear a bank advertising:
Your money will grow at a 3% annual interest rate with us.
This innocuous statement is really a differential equation. Let’s translate: A(t) will be
amount of money you have in your account at time t. On one hand, the rate at which
your money grows is the derivative dA/dt. On the other hand, we are told that this rate is
0.03A. This leads to the differential equation
dA
dt
= 0.03A.
This differential equation has a slightly different feel than the previous equation
ds
dt = 4t + 1. In the earlier example, the rate of change depends only on the independent
variable t, and we may find s(t) by integrating the velocity 4t + 1. In the banking example,
however, the rate of change depends on the dependent variable A, so we’ll need some new
techniques in order to find A(t).
Activity 7.1.
Express the following statements as differential equations. In each case, you will need
to introduce notation to describe the important quantities in the statement so be sure
to clearly state what your notation means.
(a) The population of a town grows continuously at an annual rate of 1.25%.
(b) A radioactive sample loses 5.6% of its mass every day.
(c) You have a bank account that continuously earns 4% interest every year. At the
same time, you withdraw money continually from the account at the rate of
$1000 per year.
(d) A cup of hot chocolate is sitting in a 70 ◦ room. The temperature of the
hot chocolate cools continuously by 10% of the difference between the hot
chocolate’s temperature and the room temperature every minute.
(e) A can of cold soda is sitting in a 70 ◦ room. The temperature of the soda warms
continuously at the rate of 10% of the difference between the soda’s temperature
and the room’s temperature every minute.
⊳