Prévia do material em texto

standard form
step size
threshold population
the form of a first-order linear differential equation obtained by writing the differential equation in the
form y′ + p(x)y = q(x)
the increment h that is added to the x value at each step in Euler’s Method
the minimum population that is necessary for a species to survive
KEY EQUATIONS
• Euler’s Method
xn = x0 + nh
yn = yn − 1 + h f (xn − 1, yn − 1), where h is the step size
• Separable differential equation
y′ = f (x)g(y)
• Solution concentration
du
dt = INFLOW RATE − OUTFLOW RATE
• Newton’s law of cooling
dT
dt = k(T − Ts)
• Logistic differential equation and initial-value problem
dP
dt = rP⎛
⎝1 − P
K
⎞
⎠, P(0) = P0
• Solution to the logistic differential equation/initial-value problem
P(t) = P0 Kert
⎛
⎝K − P0
⎞
⎠ + P0 ert
• Threshold population model
dP
dt = −rP⎛
⎝1 − P
K
⎞
⎠
⎛
⎝1 − P
T
⎞
⎠
• standard form
y′ + p(x)y = q(x)
• integrating factor
µ(x) = e
∫ p(x)dx
KEY CONCEPTS
4.1 Basics of Differential Equations
• A differential equation is an equation involving a function y = f (x) and one or more of its derivatives. A solution
is a function y = f (x) that satisfies the differential equation when f and its derivatives are substituted into the
equation.
• The order of a differential equation is the highest order of any derivative of the unknown function that appears in
the equation.
• A differential equation coupled with an initial value is called an initial-value problem. To solve an initial-value
problem, first find the general solution to the differential equation, then determine the value of the constant. Initial-
value problems have many applications in science and engineering.
4.2 Direction Fields and Numerical Methods
• A direction field is a mathematical object used to graphically represent solutions to a first-order differential
Chapter 4 | Introduction to Differential Equations 423
equation.
• Euler’s Method is a numerical technique that can be used to approximate solutions to a differential equation.
4.3 Separable Equations
• A separable differential equation is any equation that can be written in the form y′ = f (x)g(y).
• The method of separation of variables is used to find the general solution to a separable differential equation.
4.4 The Logistic Equation
• When studying population functions, different assumptions—such as exponential growth, logistic growth, or
threshold population—lead to different rates of growth.
• The logistic differential equation incorporates the concept of a carrying capacity. This value is a limiting value on
the population for any given environment.
• The logistic differential equation can be solved for any positive growth rate, initial population, and carrying
capacity.
4.5 First-order Linear Equations
• Any first-order linear differential equation can be written in the form y′ + p(x)y = q(x).
• We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may
not include an initial value.
• Applications of first-order linear differential equations include determining motion of a rising or falling object with
air resistance and finding current in an electrical circuit.
CHAPTER 4 REVIEW EXERCISES
True or False? Justify your answer with a proof or a
counterexample.
262. The differential equation y′ = 3x2 y − cos(x)y″ is
linear.
263. The differential equation y′ = x − y is separable.
264. You can explicitly solve all first-order differential
equations by separation or by the method of integrating
factors.
265. You can determine the behavior of all first-order
differential equations using directional fields or Euler’s
method.
For the following problems, find the general solution to the
differential equations.
266. y′ = x2 + 3ex − 2x
267. y′ = 2x + cos−1 x
268. y′ = y⎛
⎝x2 + 1⎞
⎠
269. y′ = e−y sinx
270. y′ = 3x − 2y
271. y′ = y lny
For the following problems, find the solution to the initial
value problem.
272. y′ = 8x − lnx − 3x4, y(1) = 5
273. y′ = 3x − cosx + 2, y(0) = 4
274. xy′ = y(x − 2), y(1) = 3
275. y′ = 3y2 (x + cosx), y(0) = −2
276. (x − 1)y′ = y − 2, y(0) = 0
277. y′ = 3y − x + 6x2, y(0) = −1
For the following problems, draw the directional field
424 Chapter 4 | Introduction to Differential Equations
This OpenStax book is available for free at http://cnx.org/content/col11965/1.2
associated with the differential equation, then solve the
differential equation. Draw a sample solution on the
directional field.
278. y′ = 2y − y2
279. y′ = 1
x + lnx − y, for x > 0
For the following problems, use Euler’s Method with
n = 5 steps over the interval t = [0, 1]. Then solve the
initial-value problem exactly. How close is your Euler’s
Method estimate?
280. y′ = −4yx, y(0) = 1
281. y′ = 3x − 2y, y(0) = 0
For the following problems, set up and solve the differential
equations.
282. A car drives along a freeway, accelerating according
to a = 5sin(πt), where t represents time in minutes.
Find the velocity at any time t, assuming the car starts
with an initial speed of 60 mph.
283. You throw a ball of mass 2 kilograms into the air
with an upward velocity of 8 m/s. Find exactly the time the
ball will remain in the air, assuming that gravity is given by
g = 9.8 m/s2.
284. You drop a ball with a mass of 5 kilograms out an
airplane window at a height of 5000 m. How long does it
take for the ball to reach the ground?
285. You drop the same ball of mass 5 kilograms out
of the same airplane window at the same height, except
this time you assume a drag force proportional to the ball’s
velocity, using a proportionality constant of 3 and the ball
reaches terminal velocity. Solve for the distance fallen as a
function of time. How long does it take the ball to reach the
ground?
286. A drug is administered to a patient every 24 hours
and is cleared at a rate proportional to the amount of drug
left in the body, with proportionality constant 0.2. If the
patient needs a baseline level of 5 mg to be in the
bloodstream at all times, how large should the dose be?
287. A 1000 -liter tank contains pure water and a solution
of 0.2 kg salt/L is pumped into the tank at a rate of 1 L/
min and is drained at the same rate. Solve for total amount
of salt in the tank at time t.
288. You boil water to make tea. When you pour the
water into your teapot, the temperature is 100°C. After 5
minutes in your 15°C room, the temperature of the tea is
85°C. Solve the equation to determine the temperatures of
the tea at time t. How long must you wait until the tea is at
a drinkable temperature (72°C)?
289. The human population (in thousands) of Nevada in
1950 was roughly 160. If the carrying capacity is
estimated at 10 million individuals, and assuming a
growth rate of 2% per year, develop a logistic growth
model and solve for the population in Nevada at any time
(use 1950 as time = 0). What population does your model
predict for 2000? How close is your prediction to the true
value of 1,998,257?
290. Repeat the previous problem but use Gompertz
growth model. Which is more accurate?
Chapter 4 | Introduction to Differential Equations 425
426 Chapter 4 | Introduction to Differential Equations
This OpenStax book is available for free at http://cnx.org/content/col11965/1.2
5 | SEQUENCES AND
SERIES
Figure 5.1 The Koch snowflake is constructed by using an iterative process. Starting with an equilateral triangle, at each step
of the process the middle third of each line segment is removed and replaced with an equilateral triangle pointing outward.
Chapter Outline
5.1 Sequences
5.2 Infinite Series
5.3 The Divergence and Integral Tests
5.4 Comparison Tests
5.5 Alternating Series
5.6 Ratio and Root Tests
Introduction
The Koch snowflake is constructed from an infinite number of nonoverlapping equilateral triangles. Consequently, we can
express its area as a sum of infinitely many terms. How do we add an infinite number of terms? Can a sum of an infinite
number of terms be finite? To answerthese questions, we need to introduce the concept of an infinite series, a sum with
infinitely many terms. Having defined the necessary tools, we will be able to calculate the area of the Koch snowflake (see
Example 5.8).
The topic of infinite series may seem unrelated to differential and integral calculus. In fact, an infinite series whose terms
involve powers of a variable is a powerful tool that we can use to express functions as “infinite polynomials.” We can
use infinite series to evaluate complicated functions, approximate definite integrals, and create new functions. In addition,
infinite series are used to solve differential equations that model physical behavior, from tiny electronic circuits to Earth-
orbiting satellites.
5.1 | Sequences
Learning Objectives
5.1.1 Find the formula for the general term of a sequence.
5.1.2 Calculate the limit of a sequence if it exists.
5.1.3 Determine the convergence or divergence of a given sequence.
Chapter 5 | Sequences and Series 427
	Chapter 5. Sequences and Series
	5.1. Sequences*