Calculus Early Transcendentals, Binder Ready Version
Calculus Early Transcendentals, Binder Ready Version
11th Edition
ISBN: 9781118883822
Author: Howard Anton, Irl C. Bivens, Stephen Davis
Publisher: WILEY
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Differential Equations
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Chapter 8, Problem 24RE

(a) Solve the initial-value problem

y ' − y = x sin 3 x ,   y ( 0 ) = 1

by the method of integrating factors, using a CAS to perform any difficult integrations.

(b) Use the CAS to solve the initial-value problem directly, and confirm that the answer to consistent with that obtained in part(a).

(c) Graph the solution.

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1) Consider the equation y'+2x y-y'=-2x, x>0. (a) Determine the values of the constants k and r such that y = kx' is a solution to the equation. (b) Use the result from part (a) to determine the general solution to the equation.
(d) Solve the equation (IVP) (1+ x²) – x(1+ 2y)y' = 0 y(1) = 0
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Chapter 8 Solutions

Calculus Early Transcendentals, Binder Ready Version

Ch. 8.1 - Match each differential equation with its family...Ch. 8.1 - If y=C1e2x+C2xe2x is the general solution of a...Ch. 8.1 - The graph of a differentiable function y=yx passes...Ch. 8.1 - A glass of ice water with a temperature of 36F is...Ch. 8.1 - Confirm that y=3ex3 is a solution of the...Ch. 8.1 - Confirm that y=14x4+2cosx+1 is a solution of the...Ch. 8.1 - State the order of the differential equation, and...Ch. 8.1 - State the order of the differential equation, and...Ch. 8.1 - Determine whether the statement is true or false....Ch. 8.1 - Determine whether the statement is true or false....
Ch. 8.1 - Determine whether the statement is true or false....Ch. 8.1 - Determine whether the statement is true or false....Ch. 8.1 - In each part, verify that the functions are...Ch. 8.1 - In each part, verify that the functions are...Ch. 8.1 - In each part, verify that the functions are...Ch. 8.1 - In each part, verify that the functions are...Ch. 8.1 - In each part, verify that the functions are...Ch. 8.1 - In each part, verify that the functions are...Ch. 8.1 - Use the results of Exercises 9-14 to find a...Ch. 8.1 - Use the results of Exercises 9-14 to find a...Ch. 8.1 - Use the results of Exercises 9-14 to find a...Ch. 8.1 - Use the results of Exercises 9-14 to find a...Ch. 8.1 - Use the results of Exercises 9-14 to find a...Ch. 8.1 - Use the results of Exercises 9-14 to find a...Ch. 8.1 - Find a solution to the initial-value problem....Ch. 8.1 - Find a solution to the initial-value problem. Ch. 8.1 - Find a solution to the initial-value problem. Ch. 8.1 - Find a solution to the initial-value problem....Ch. 8.1 - Find a solution to the initial-value problem. Ch. 8.1 - Find a solution to the initial-value problem....Ch. 8.1 - (a) Suppose that a quantity y=yt increases at a...Ch. 8.1 - (a) Suppose that a quantity y=yt changes in such a...Ch. 8.1 - (a) Suppose that a particle moves along an s-axis...Ch. 8.1 - Suppose that a body moves along an s-axis through...Ch. 8.1 - Consider a solution y=yt to the uninhibited...Ch. 8.1 - Consider the logistic model for population growth....Ch. 8.1 - Consider the model for the spread of disease. (a)...Ch. 8.1 - Explain why there is exactly one constant solution...Ch. 8.1 - Show that if c1 and c2 axe any constants, the...Ch. 8.1 - (a) Use the result of Exercise 35 to solve the...Ch. 8.2 - Solve the first-order separable equation...Ch. 8.2 - Suppose that a quantity y=yt has an exponential...Ch. 8.2 - Suppose that a quantity y=yt has an exponential...Ch. 8.2 - The initial-value problem dydx=xy,y(0)=1 has...Ch. 8.2 - Solve the differential equation by separation of...Ch. 8.2 - Solve the differential equation by separation of...Ch. 8.2 - Solve the differential equation by separation of...Ch. 8.2 - Solve the differential equation by separation of...Ch. 8.2 - Solve the differential equation by separation of...Ch. 8.2 - Solve the differential equation by separation of...Ch. 8.2 - Solve the differential equation by separation of...Ch. 8.2 - Solve the differential equation by separation of...Ch. 8.2 - Solve the differential equation by separation of...Ch. 8.2 - Solve the differential equation by separation of...Ch. 8.2 - Solve the initial-value problem by separation of...Ch. 8.2 - Solve the initial-value problem by separation of...Ch. 8.2 - Solve the initial-value problem by separation of...Ch. 8.2 - Solve the initial-value problem by separation of...Ch. 8.2 - (a) Sketch some typical integral curves of the...Ch. 8.2 - (a) Sketch some typical integral curves of the...Ch. 8.2 - Solve the differential equation and then use a...Ch. 8.2 - Solve the differential equation and then use a...Ch. 8.2 - Solve the differential equation. If you have a CAS...Ch. 8.2 - Solve the differential equation. If you have a CAS...Ch. 8.2 - Determine whether the statement is true or false....Ch. 8.2 - Determine whether the statement is true or false....Ch. 8.2 - Determine whether the statement is true or false....Ch. 8.2 - Determine whether the statement is true or false....Ch. 8.2 - Suppose that the initial condition in Example 1...Ch. 8.2 - Find all ordered pairs (x0,y0) such that if the...Ch. 8.2 - Find an equation of a curve with x-intercept 2...Ch. 8.2 - Use a graphing utility to generate a curve that...Ch. 8.2 - Suppose that an initial population of 10,000...Ch. 8.2 - A cell of the bacterium E. coli divides into two...Ch. 8.2 - Radon-222 is a radioactive gas with a half-life of...Ch. 8.2 - Methylmercury is a toxic compound that can lead to...Ch. 8.2 - Suppose that 100 fruit flies are placed in a...Ch. 8.2 - Suppose that the town of Grayrock had a population...Ch. 8.2 - A scientist wants to determine the half-life of a...Ch. 8.2 - Suppose that 30 of a certain radioactive substance...Ch. 8.2 - Prob. 37ESCh. 8.2 - (a) What effect does increasing y0 and keeping k...Ch. 8.2 - (a) There is a trick, called the Rule of 70, that...Ch. 8.2 - Find a formula for the tripling time of an...Ch. 8.2 - In 1950, a research team digging near Folsom, New...Ch. 8.2 - (a) Use a graphing utility to make a graph of prem...Ch. 8.2 - (a) It is currently accepted that the half-life of...Ch. 8.2 - Suppose that a quantity y has an exponential...Ch. 8.2 - (a) Show that if a quantity y=yt has an...Ch. 8.2 - Suppose that P dollars is invested at an annual...Ch. 8.2 - (a) If 1000 is invested at 8 per year compounded...Ch. 8.2 - What is the effective annual interest rate for an...Ch. 8.2 - Assume that y=yt satisfies the logistic equation...Ch. 8.2 - Use your answer to Exercise 49 to derive a...Ch. 8.2 - The graph of a solution to the logistic equation...Ch. 8.2 - The graph of a logistic model y=y0Ly0+Ly0ekt is...Ch. 8.2 - The graph of a logistic model y=y0Ly0+Ly0ekt is...Ch. 8.2 - Plot a solution to the initial-value problem...Ch. 8.2 - Suppose that the growth of a population y=yt is...Ch. 8.2 - Suppose that the growth of a population y=yt is...Ch. 8.2 - Suppose that a university residence hall houses...Ch. 8.2 - Suppose that at time t=0 an object with...Ch. 8.2 - A cup of water with a temperature of 95C is placed...Ch. 8.2 - A glass of lemonade with a temperature of 40F is...Ch. 8.2 - A rocket, fired upward from rest at time t=0, has...Ch. 8.2 - A bullet of mass m, fired straight up with an...Ch. 8.2 - Suppose that a tank containing a liquid is vented...Ch. 8.2 - Suppose that a tank containing a liquid is vented...Ch. 8.2 - Suppose that a particle moving along the x-axis...Ch. 8.2 - Suppose that a particle moving along the x-axis...Ch. 8.2 - Use implicit differentiation to prove that any...Ch. 8.2 - Prove that a solution to the initial-value problem...Ch. 8.2 - Let L denote a tangent line at x,y to a solution...Ch. 8.3 - Match each differential equation with its slop...Ch. 8.3 - The slope field for y=y/x at the 16 gridpoints...Ch. 8.3 - When using Euler's Method on the initial-value...Ch. 8.3 - Consider the initial-value problem y=y,y0=1. (a)...Ch. 8.3 - Sketch the slope field for y=xy/4 at the 25...Ch. 8.3 - Sketch the slope field for y+y=2 at the 25...Ch. 8.3 - A slope field for the differential equation y=1y...Ch. 8.3 - Solve the initial-value problems in Exercise 3,...Ch. 8.3 - Use the slope field in Exercise 3 to make a...Ch. 8.3 - In parts (a)-(f), match the differential equation...Ch. 8.3 - Use Euler's Method with the given step size xort...Ch. 8.3 - Use Euler's Method with the given step size xort...Ch. 8.3 - Use Euler's Method with the given step size xort...Ch. 8.3 - Use Euler's Method with the given step size xort...Ch. 8.3 - Consider the initial-value problem y=sint,y0=0 Use...Ch. 8.3 - Determine whether the statement is true or false....Ch. 8.3 - Determine whether the statement is true or false....Ch. 8.3 - Determine whether the statement is true or false....Ch. 8.3 - Determine whether the statement is true or false....Ch. 8.3 - (a) Show that the solution of the initial-value...Ch. 8.3 - The accompanying figure shows a slope field for...Ch. 8.3 - Refer to slope field II in Quick Check Exercise 1....Ch. 8.3 - Refer to the slope field in Exercise 3 and...Ch. 8.3 - Consider the initial-value problem dydx=y2,y0=1...Ch. 8.3 - A slope field of the form y=fy is said to be...Ch. 8.3 - (a) Solve the equation y=y and show that every...Ch. 8.3 - (a) Find a slope field whose integral curve...Ch. 8.3 - (a) Find a slope field whose integral curve...Ch. 8.3 - Consider the initial-value problem y=y,y0=1, and...Ch. 8.3 - Explain the connection between Euler's Method and...Ch. 8.4 - Solve the first-order linear differential equation...Ch. 8.4 - An integrating factor for dydx+yx=q(x) is.Ch. 8.4 - At time t=0, a tank contains 30 oz of salt of salt...Ch. 8.4 - Solve the differential equation by the method of...Ch. 8.4 - Solve the differential equation by the method of...Ch. 8.4 - Solve the differential equation by the method of...Ch. 8.4 - Solve the differential equation by the method of...Ch. 8.4 - Solve the differential equation by the method of...Ch. 8.4 - Solve the differential equation by the method of...Ch. 8.4 - Solve the initial-value problem. Ch. 8.4 - Solve the initial-value problem. Ch. 8.4 - Solve the initial-value problem. Ch. 8.4 - Solve the initial-value problem. dydt+y=2,y(0)=1Ch. 8.4 - Determine whether the statement is true or false....Ch. 8.4 - Determine whether the statement is true or false....Ch. 8.4 - Determine whether the statement is true or false....Ch. 8.4 - Determine whether the statement is true or false....Ch. 8.4 - A slope field for the differential equation y=2yx...Ch. 8.4 - Solve the initial-value problem is Exercise 15,...Ch. 8.4 - Use the slope fields in Exercise 15 to make a...Ch. 8.4 - Consider the slope field in Exercise 15. (a) Use...Ch. 8.4 - (a) Use Euler’s Method with a step size of x=0.2...Ch. 8.4 - It was stated at the end of Section 8.3 that...Ch. 8.4 - At time a tank contains 25 oz of salt dissolved...Ch. 8.4 - A tank initially contains 200 gal of pure water....Ch. 8.4 - Fish is a rich source of protein and omega-3 fatty...Ch. 8.4 - The concentration of methylmercury in “light�...Ch. 8.4 - (a) Use the method of integrating factors to...Ch. 8.4 - Suppose a fully equipped skydiver weighing 240 lb...Ch. 8.4 - The accompanying figure is a schematic diagram of...Ch. 8.4 - Find l(t) for the electrical circuit in Exercise...Ch. 8.4 - (a) Prove that any function y=y(x) defined by...Ch. 8.4 - Prove that solutions need not be unique for...Ch. 8.4 - Explain why the quantity n the Method of...Ch. 8 - Classify the following first-order differential...Ch. 8 - Which of the given differential equations are...Ch. 8 - Solve the differential equation by the method of...Ch. 8 - Solve the differential equation by the method of...Ch. 8 - Solve the differential equation by method of...Ch. 8 - Solve the initial-value problem by the method of...Ch. 8 - Solve the initial-value problem by the method of...Ch. 8 - Solve the initial-value problem by the method of...Ch. 8 - Sketch the integral curve of y=2xy2 that passes...Ch. 8 - Sketch the integral curve of 2yy=1 that passes...Ch. 8 - Sketch the slope fields for y=xy/8 at the 25 grid...Ch. 8 - Solve the differential equation y=xy/8 , and find...Ch. 8 - Use Euler’s Method with the given step size x to...Ch. 8 - Use Euler’s Method with the given step size x to...Ch. 8 - Consider the initial-value problem y=cos2t,y(0)=1...Ch. 8 - Cloth found in an Egyptian pyramid contains 78.5...Ch. 8 - In each part, find an exponential growth model...Ch. 8 - Suppose that an initial population of 5000...Ch. 8 - Solve the differential equation by the method of...Ch. 8 - Solve the differential equation by the method of...Ch. 8 - Solve the initial-value problem by the method of...Ch. 8 - Solve the initial-value problem by the method of...Ch. 8 - Solve the initial-value problem by the method of...Ch. 8 - (a) Solve the initial-value problem...Ch. 8 - A tank contains 1000 gal of fresh water. At time...Ch. 8 - Suppose that a room containing 1200ft3 of air is...

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