7.5. MODELING WITH DIFFERENTIAL EQUATIONS
427
7.5 Modeling with differential equations
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How can we use differential equations to describe phenomena in the world around
us?
• How can we use differential equations to better understand these phenomena?
Introduction
In our work to date, we have seen several ways that differential equations arise in the
natural world, from the growth of a population to the temperature of a cup of coffee. In
this section, we will look more closely at how differential equations give us a natural way to
describe various phenoma. As we’ll see, the key is to focus on understanding the different
factors that cause a quantity to change.
Preview Activity 7.5. Any time that the rate of change of a quantity is related to the
amount of a quantity, a differential equation naturally arises. In the following two problems,
we see two such scenarios; for each, we want to develop a differential equation whose
solution is the quantity of interest.
(a) Suppose you have a bank account in which money grows at an annual rate of 3%.
(i) If you have $10,000 in the account, at what rate is your money growing?
(ii) Suppose that you are also withdrawing money from the account at $1,000
per year. What is the rate of change in the amount of money in the account?
What are the units on this rate of change?
(b) Suppose that a water tank holds 100 gallons and that a salty solution, which
contains 20 grams of salt in every gallon, enters the tank at 2 gallons per minute.
(i) How much salt enters the tank each minute?
(ii) Suppose that initially there are 300 grams of salt in the tank. How much salt
is in each gallon at this point in time?
(iii) Finally, suppose that evenly mixed solution is pumped out of the tank at the
rate of 2 gallons per minute. How much salt leaves the tank each minute?
(iv) What is the total rate of change in the amount of salt in the tank?
⊲⊳
427
7.5 Modeling with differential equations
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How can we use differential equations to describe phenomena in the world around
us?
• How can we use differential equations to better understand these phenomena?
Introduction
In our work to date, we have seen several ways that differential equations arise in the
natural world, from the growth of a population to the temperature of a cup of coffee. In
this section, we will look more closely at how differential equations give us a natural way to
describe various phenoma. As we’ll see, the key is to focus on understanding the different
factors that cause a quantity to change.
Preview Activity 7.5. Any time that the rate of change of a quantity is related to the
amount of a quantity, a differential equation naturally arises. In the following two problems,
we see two such scenarios; for each, we want to develop a differential equation whose
solution is the quantity of interest.
(a) Suppose you have a bank account in which money grows at an annual rate of 3%.
(i) If you have $10,000 in the account, at what rate is your money growing?
(ii) Suppose that you are also withdrawing money from the account at $1,000
per year. What is the rate of change in the amount of money in the account?
What are the units on this rate of change?
(b) Suppose that a water tank holds 100 gallons and that a salty solution, which
contains 20 grams of salt in every gallon, enters the tank at 2 gallons per minute.
(i) How much salt enters the tank each minute?
(ii) Suppose that initially there are 300 grams of salt in the tank. How much salt
is in each gallon at this point in time?
(iii) Finally, suppose that evenly mixed solution is pumped out of the tank at the
rate of 2 gallons per minute. How much salt leaves the tank each minute?
(iv) What is the total rate of change in the amount of salt in the tank?
⊲⊳