Differential Equations
4th Edition
ISBN: 9780495561989
Author: Paul Blanchard, Robert L. Devaney, Glen R. Hall
Publisher: Cengage Learning
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Chapter 1.4, Problem 7E
To determine
To calculate: the approximate values by Euler’s method with the given step size
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The following table indicates the unit prices (in Rands) and quantities of three consumer products
to be held in a supermarket warehouse in Lenasia over the time period from April to July 2025.
APRIL 2025
JULY 2025
PRODUCT
Unit Price (po)
Quantity (q0))
Unit Price (p₁)
Quantity (q1)
Mineral Water
R23.70
403
R25.70
423
H&S Shampoo
R77.00
922
R79.40
899
Toilet Paper
R106.50
725
R104.70
730
The Independent Institute of Education (Pty) Ltd 2025
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The following table indicates the unit prices (in Rands) and quantities of three meals sold every year
by a small restaurant over the years 2023 and 2025.
2023
2025
MEAL
Unit Price (po) Quantity (q0))
Unit Price (P₁)
Quantity (q₁)
Lasagne
R125
1055
R145
1125
Pizza
R110
2115
R130
2195
Pasta
R95
1950
R120
2250
Q.2.1
Using 2023 as the base year, compute the individual price relatives in 2025 for (10)
lasagne and pasta. Interpret each of your answers.
0.2.2
Using 2023 as the base year, compute the Laspeyres price index for all of the meals (8)
for 2025. Interpret your answer.
Q.2.3
Using 2023 as the base year, compute the Paasche price index for all of the meals (7)
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Chapter 1 Solutions
Differential Equations
Ch. 1.1 - In Exercises 1 and 2, find the equilibrium...Ch. 1.1 - In Exercises 1 and 2, find the equilibrium...Ch. 1.1 - Consider the population model dPdt=0.4P(1P230)...Ch. 1.1 - Consider the population model ...Ch. 1.1 - Consider the differential equation dydt=y3y212y...Ch. 1.1 - In Exercises 6—10, we consider the phenomenon of...Ch. 1.1 - In Exercises 6—10, we consider the phenomenon of...Ch. 1.1 - In Exercises 6—10, we consider the phenomenon of...Ch. 1.1 - In Exercises 6—10, we consider the phenomenon of...Ch. 1.1 - In Exercises 6—10, we consider the phenomenon of...
Ch. 1.1 - MacQuarie Island is a small island about half-way...Ch. 1.1 - The velocity u of a freefalling skydiver is well...Ch. 1.1 - Exercises 13—15 consider an elementary model of...Ch. 1.1 - Exercises 13—15 consider an elementary model of...Ch. 1.1 - Exercises 13—15 consider an elementary model of...Ch. 1.1 - The expenditure on education in the U.S. is given...Ch. 1.1 - Suppose a species of fish in a particular lake has...Ch. 1.1 - Suppose that the growth-rate parameter k = 0.3 and...Ch. 1.1 - The rhinoceros is now extremely rare. Suppose...Ch. 1.1 - While it is difficult to imagine a time before...Ch. 1.1 - For the following predator-prey systems, identify...Ch. 1.1 - In the following predator-prey population models,x...Ch. 1.1 - The following systems are models of the...Ch. 1.2 - Bob. Glen. and Paul are once again sitting around...Ch. 1.2 - Make up a differential equation of the form ...Ch. 1.2 - Make up a differential equation of the form dy/dt...Ch. 1.2 - In Section 1.1, we guessed solutions to the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - Prob. 32ECh. 1.2 - Prob. 33ECh. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - Prob. 38ECh. 1.2 - A 5-gallon bucket is full of pure water. Suppose...Ch. 1.2 - Consider the following very simple model of blood...Ch. 1.2 - A cup of hot chocolate is initially 170o Fand is...Ch. 1.2 - Suppose you are having a dinner party for a large...Ch. 1.2 - Prob. 43ECh. 1.3 - In Exercises 1-6, sketch the slope fields for the...Ch. 1.3 - Prob. 2ECh. 1.3 - Prob. 3ECh. 1.3 - Prob. 4ECh. 1.3 - Prob. 5ECh. 1.3 - Prob. 6ECh. 1.3 - In Exercises 710, a differential equation and its...Ch. 1.3 - Prob. 8ECh. 1.3 - Prob. 9ECh. 1.3 - Prob. 10ECh. 1.3 - Suppose we know that the function f(t, y) is...Ch. 1.3 - Prob. 12ECh. 1.3 - Prob. 13ECh. 1.3 - Prob. 14ECh. 1.3 - Consider the autonomous differential equation ...Ch. 1.3 - Eight differential equations and four slope fields...Ch. 1.3 - Prob. 17ECh. 1.3 - Prob. 18ECh. 1.3 - Prob. 19ECh. 1.3 - Prob. 20ECh. 1.3 - Prob. 21ECh. 1.3 - Prob. 22ECh. 1.4 - In Exercises 14, use EulersMethod to perform...Ch. 1.4 - In Exercises 14, use EulersMethod to perform...Ch. 1.4 - Prob. 3ECh. 1.4 - In Exercises 14, use EulersMethod to perform...Ch. 1.4 - In Exercises 510, use Euler’s method with the...Ch. 1.4 - In Exercises 510, use Euler’s method with the...Ch. 1.4 - Prob. 7ECh. 1.4 - Prob. 8ECh. 1.4 - Prob. 9ECh. 1.4 - Prob. 10ECh. 1.4 - Prob. 11ECh. 1.4 - Prob. 12ECh. 1.4 - Prob. 13ECh. 1.4 - Prob. 14ECh. 1.4 - Consider the initial-value problem dy/dt= y ,y(0)...Ch. 1.4 - Consider the initial-value problem dy/dt= 2 ...Ch. 1.4 - As we saw in Exercise 19 of Section 1.3, the...Ch. 1.4 - Prob. 18ECh. 1.4 - Prob. 19ECh. 1.4 - Prob. 20ECh. 1.4 - Prob. 21ECh. 1.5 - In Exercises 1—4, we refer to a function f, but we...Ch. 1.5 - Prob. 2ECh. 1.5 - In Exercises 1—4, we refer to a function f, but we...Ch. 1.5 - In Exercises 1—4, we refer to a function f, but we...Ch. 1.5 - Prob. 5ECh. 1.5 - In Exercises 5—8, an initial condition for the...Ch. 1.5 - Prob. 7ECh. 1.5 - In Exercises 5—8, an initial condition for the...Ch. 1.5 - (a) Show that y1(t)=t2 and y2(t)=t2+1 are...Ch. 1.5 - Consider the differential equation dy/dt=2y (a)...Ch. 1.5 - Consider the differential equation dydt=yt2 (a)...Ch. 1.5 - (a) Show that y1(t)=1t1 and y2(t)=1t2 are...Ch. 1.5 - Prob. 13ECh. 1.5 - In Exercises 13—16, an initial-value problem is...Ch. 1.5 - In Exercises 13—16, an initial-value problem is...Ch. 1.5 - In Exercises 13—16, an initial-value problem is...Ch. 1.5 - Prob. 17ECh. 1.5 - We have emphasized that the Uniqueness Theorem...Ch. 1.6 - In Exercises 112, sketch the phase lines for the...Ch. 1.6 - In Exercises 112, sketch the phase lines for the...Ch. 1.6 - Prob. 3ECh. 1.6 - In Exercises 112, sketch the phase lines for the...Ch. 1.6 - Prob. 5ECh. 1.6 - In Exercises 112, sketch the phase lines for the...Ch. 1.6 - In Exercises 112, sketch the phase lines for the...Ch. 1.6 - In Exercises 112, sketch the phase lines for the...Ch. 1.6 - Prob. 9ECh. 1.6 - In Exercises 112, sketch the phase lines for the...Ch. 1.6 - Prob. 11ECh. 1.6 - Prob. 12ECh. 1.6 - In Exercises 1321, a differential equation and...Ch. 1.6 - In Exercises 1321, a differential equation and...Ch. 1.6 - In Exercises 1321, a differential equation and...Ch. 1.6 - Prob. 16ECh. 1.6 - ]In Exercises 1321, a differential equation and...Ch. 1.6 - Prob. 18ECh. 1.6 - In Exercises 1321, a differential equation and...Ch. 1.6 - Prob. 20ECh. 1.6 - Prob. 21ECh. 1.6 - In Exercises 2227, describe the long-term behavior...Ch. 1.6 - Prob. 23ECh. 1.6 - In Exercises 2227, describe the long-term behavior...Ch. 1.6 - In Exercises 2227, describe the long-term behavior...Ch. 1.6 - In Exercises 2227, describe the long-term behavior...Ch. 1.6 - Prob. 27ECh. 1.6 - Prob. 28ECh. 1.6 - Prob. 29ECh. 1.6 - In Exercises 2932, the graph of a function f(y) is...Ch. 1.6 - In Exercises 2932, the graph of a function f(y) is...Ch. 1.6 - In Exercises 2932, the graph of a function f(y) is...Ch. 1.6 - Prob. 33ECh. 1.6 - Prob. 34ECh. 1.6 - Prob. 35ECh. 1.6 - Prob. 36ECh. 1.6 - Eight differential equations and four phase lines...Ch. 1.6 - Prob. 38ECh. 1.6 - Prob. 39ECh. 1.6 - Consider the Ermentrout-Kopell model for the...Ch. 1.6 - Prob. 41ECh. 1.6 - Prob. 42ECh. 1.6 - Prob. 43ECh. 1.6 - Prob. 44ECh. 1.6 - Let x(t) be the amount of time between two...Ch. 1.6 - Prob. 46ECh. 1.6 - Use the model in Exercise 45 to predict what...Ch. 1.6 - Prob. 48ECh. 1.7 - In Exercises 1-6, locate the bifurcation values...Ch. 1.7 - In Exercises 1-6, locate the bifurcation values...Ch. 1.7 - In Exercises 1-6, locate the bifurcation values...Ch. 1.7 - In Exercises 1-6, locate the bifurcation values...Ch. 1.7 - In Exercises 1-6, locate the bifurcation values...Ch. 1.7 - Prob. 6ECh. 1.7 - Prob. 7ECh. 1.7 - Prob. 8ECh. 1.7 - Prob. 9ECh. 1.7 - Prob. 10ECh. 1.7 - The graph to the right is the graph of a function...Ch. 1.7 - The graph to the right is the graph of a function...Ch. 1.7 - Six one-parameter families of different equations...Ch. 1.7 - Consider the Ermentrout-Kopell model for the...Ch. 1.7 - Prob. 15ECh. 1.7 - Sketch the graph of a function g(y) such that the...Ch. 1.7 - Is it possible to find a continuous function...Ch. 1.7 - Prob. 18ECh. 1.7 - Consider the population model dPdt=2PP250 for a...Ch. 1.7 - Prob. 20ECh. 1.7 - Prob. 21ECh. 1.7 - Prob. 22ECh. 1.7 - Prob. 23ECh. 1.8 - In Exercises 1-6, find the general solution of the...Ch. 1.8 - Prob. 2ECh. 1.8 - In Exercises 1-6, find the general solution of the...Ch. 1.8 - Prob. 4ECh. 1.8 - In Exercises 1-6, find the general solution of the...Ch. 1.8 - In Exercises 1-6, find the general solution of the...Ch. 1.8 - In Exercises 7-12, solve the given initial-value...Ch. 1.8 - In Exercises 7-12, solve the given initial-value...Ch. 1.8 - In Exercises 7-12, solve the given initial-value...Ch. 1.8 - In Exercises 7-12, solve the given initial-value...Ch. 1.8 - In Exercises 7-12, solve the given initial-value...Ch. 1.8 - In Exercises 7-12, solve the given initial-value...Ch. 1.8 - Consider the nonhomogeneous linear equation...Ch. 1.8 - Prob. 14ECh. 1.8 - Prob. 15ECh. 1.8 - Prob. 16ECh. 1.8 - Consider the nonlinear differential equation...Ch. 1.8 - Prob. 18ECh. 1.8 - Prob. 19ECh. 1.8 - Consider the nonhomogeneous linear equation...Ch. 1.8 - In Exercises 21-24, find the general solution and...Ch. 1.8 - In Exercises 21-24, find the general solution and...Ch. 1.8 - In Exercises 21-24, find the general solution and...Ch. 1.8 - In Exercises 21-24, find the general solution and...Ch. 1.8 - Prob. 25ECh. 1.8 - Prob. 26ECh. 1.8 - In Exercises 25-28, give a brief qualitative...Ch. 1.8 - Prob. 28ECh. 1.8 - A person initially places $1,000 in a savings...Ch. 1.8 - A student has saved $70,000 for her college...Ch. 1.8 - A college professor contributes $5,000 per year...Ch. 1.8 - Prob. 32ECh. 1.8 - Prob. 33ECh. 1.8 - Prob. 34ECh. 1.9 - In Exercises 1-6, find the general solution of the...Ch. 1.9 - In Exercises 1-6, find the general solution of the...Ch. 1.9 - In Exercises 1-6, find the general solution of the...Ch. 1.9 - In Exercises 1-6, find the general solution of the...Ch. 1.9 - In Exercises 1-6, find the general solution of the...Ch. 1.9 - In Exercises 1-6, find the general solution of the...Ch. 1.9 - In Exercises 7-12, solve the given initial-value...Ch. 1.9 - In Exercises 7-12, solve the given initial-value...Ch. 1.9 - Prob. 9ECh. 1.9 - Prob. 10ECh. 1.9 - Prob. 11ECh. 1.9 - In Exercises 7-12, solve the given initial-value...Ch. 1.9 - In Exercises 13-18, the differential equation is...Ch. 1.9 - Prob. 14ECh. 1.9 - Prob. 15ECh. 1.9 - Prob. 16ECh. 1.9 - Prob. 17ECh. 1.9 - Prob. 18ECh. 1.9 - Prob. 19ECh. 1.9 - Prob. 20ECh. 1.9 - Prob. 21ECh. 1.9 - Prob. 22ECh. 1.9 - Prob. 23ECh. 1.9 - A 30-gallon tank initially contains 15 gallons of...Ch. 1.9 - A 400-gallon tank initially contains 200 gallons...Ch. 1.9 - A 100-gallon tank initially contains loo gallons...Ch. 1.9 - Suppose a 50-gallon tank contains a volume Vo of...Ch. 1 - Short answer exercises: Exercises 1-10 focus on...Ch. 1 - Short answer exercises: Exercises 1-10 focus on...Ch. 1 - Short answer exercises: Exercises 1-10 focus on...Ch. 1 - Prob. 4RECh. 1 - Short answer exercises: Exercises 1-10 focus on...Ch. 1 - Short answer exercises: Exercises 1-10 focus on...Ch. 1 - Prob. 7RECh. 1 - Prob. 8RECh. 1 - Prob. 9RECh. 1 - Prob. 10RECh. 1 - Prob. 11RECh. 1 - Prob. 12RECh. 1 - Prob. 13RECh. 1 - Prob. 14RECh. 1 - True-false: For Exercises 11-20, determine if the...Ch. 1 - Prob. 16RECh. 1 - True-false: For Exercises 11-20, determine if the...Ch. 1 - Prob. 18RECh. 1 - Prob. 19RECh. 1 - Prob. 20RECh. 1 - Prob. 21RECh. 1 - Prob. 22RECh. 1 - Prob. 23RECh. 1 - Prob. 24RECh. 1 - Prob. 25RECh. 1 - Prob. 26RECh. 1 - Prob. 27RECh. 1 - Prob. 28RECh. 1 - In Exercises 21-29, (a) specify if the given...Ch. 1 - Prob. 30RECh. 1 - Prob. 31RECh. 1 - Prob. 32RECh. 1 - Prob. 33RECh. 1 - Prob. 34RECh. 1 - In Exercises 30-39, (a) specify if the given...Ch. 1 - Prob. 36RECh. 1 - Prob. 37RECh. 1 - Prob. 38RECh. 1 - Prob. 39RECh. 1 - Prob. 40RECh. 1 - Prob. 41RECh. 1 - Consider the autonomous differential equation...Ch. 1 - Prob. 43RECh. 1 - Prob. 44RECh. 1 - Prob. 45RECh. 1 - Prob. 46RECh. 1 - Prob. 47RECh. 1 - Prob. 48RECh. 1 - Eight differential equations and four slope fields...Ch. 1 - Prob. 50RECh. 1 - Prob. 51RECh. 1 - Consider the differential equation dy/dt=2ty2 ....Ch. 1 - Prob. 53RECh. 1 - A 1000-galIon tank initially contains a mixture of...
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