7.5. MODELING WITH DIFFERENTIAL EQUATIONS
433
Summary
In this section, we encountered the following important ideas:
• Differential equations arise in a situation when we understand how various factors
cause a quantity to change.
• We may use the tools we have developed so far—slope fields, Euler’s methods, and
our method for solving separable equations—to understand a quantity described by a
differential equation.
Exercises
1. Congratulations, you just won the lottery! In one option presented to you, you will be
paid one million dollars a year for the next 25 years. You can deposit this money in an
account that will earn 5% each year.
(a) Set up a differential equation that describes the rate of change in the amount
of money in the account. Two factors cause the amount to grow—first, you are
depositing one millon dollars per year and second, you are earning 5% interest.
(b) If there is no amount of money in the account when you open it, how much
money will you have in the account after 25 years?
(c) The second option presented to you is to take a lump sum of 10 million dollars,
which you will deposit into a similar account. How much money will you have
in that account after 25 years?
(d) Do you prefer the first or second option? Explain your thinking.
(e) At what time does the amount of money in the account under the first option
overtake the amount of money in the account under the second option?
2. When a skydiver jumps from a plane, gravity causes her downward velocity to increase
at the rate of g ≈ 9.8 meters per second squared. At the same time, wind resistance
causes her velocity to decrease at a rate proportional to the velocity.
(a) Using k to represent the constant of proportionality, write a differential equation
that describes the rate of change of the skydiver’s velocity.
(b) Find any equilibrium solutions and decide whether they are stable or unstable.
Your result should depend on k.
(c) Suppose that the initial velocity is zero. Find the velocity v(t).
(d) A typical terminal velocity for a skydiver falling face down is 54 meters per
second. What is the value of k for this skydiver?
433
Summary
In this section, we encountered the following important ideas:
• Differential equations arise in a situation when we understand how various factors
cause a quantity to change.
• We may use the tools we have developed so far—slope fields, Euler’s methods, and
our method for solving separable equations—to understand a quantity described by a
differential equation.
Exercises
1. Congratulations, you just won the lottery! In one option presented to you, you will be
paid one million dollars a year for the next 25 years. You can deposit this money in an
account that will earn 5% each year.
(a) Set up a differential equation that describes the rate of change in the amount
of money in the account. Two factors cause the amount to grow—first, you are
depositing one millon dollars per year and second, you are earning 5% interest.
(b) If there is no amount of money in the account when you open it, how much
money will you have in the account after 25 years?
(c) The second option presented to you is to take a lump sum of 10 million dollars,
which you will deposit into a similar account. How much money will you have
in that account after 25 years?
(d) Do you prefer the first or second option? Explain your thinking.
(e) At what time does the amount of money in the account under the first option
overtake the amount of money in the account under the second option?
2. When a skydiver jumps from a plane, gravity causes her downward velocity to increase
at the rate of g ≈ 9.8 meters per second squared. At the same time, wind resistance
causes her velocity to decrease at a rate proportional to the velocity.
(a) Using k to represent the constant of proportionality, write a differential equation
that describes the rate of change of the skydiver’s velocity.
(b) Find any equilibrium solutions and decide whether they are stable or unstable.
Your result should depend on k.
(c) Suppose that the initial velocity is zero. Find the velocity v(t).
(d) A typical terminal velocity for a skydiver falling face down is 54 meters per
second. What is the value of k for this skydiver?