Fundamentals of Differential Equations (9th Edition)
9th Edition
ISBN: 9780321977069
Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider
Publisher: PEARSON
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Quarterly sales of videos in the Leeds "Disney" store are shown in figure 1. Below
is the code and output for an analysis of these data in R, with the sales data stored
in the time series object X.
Explain what is being done at points (i)-(iv) in the R code. Explain what is the
difference between (v) and (vi) in the R code. Explain, giving reasons, which of
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Given that the sales for the four quarters of 2018 were 721, 935, 649, and 1071,
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Chapter 1 Solutions
Fundamentals of Differential Equations (9th Edition)
Ch. 1.1 - In Problems 112, a differential equation is given...Ch. 1.1 - In Problems 112, a differential equation is given...Ch. 1.1 - In Problems 112, a differential equation is given...Ch. 1.1 - In Problems 112, a differential equation is given...Ch. 1.1 - In Problems 112, a differential equation is given...Ch. 1.1 - In Problems 112, a differential equation is given...Ch. 1.1 - In Problems 112, a differential equation is given...Ch. 1.1 - In Problems 112, a differential equation is given...Ch. 1.1 - Prob. 9ECh. 1.1 - In Problems 112, a differential equation is given...
Ch. 1.1 - In Problems 112, a differential equation is given...Ch. 1.1 - Prob. 12ECh. 1.1 - In Problems 1316, write a differential equation...Ch. 1.1 - In Problems 1316, write a differential equation...Ch. 1.1 - In Problems 1316, write a differential equation...Ch. 1.1 - In Problems 1316, write a differential equation...Ch. 1.1 - Prob. 17ECh. 1.2 - (a) Show that (x) = x2 is an explicit solution to...Ch. 1.2 - (a) Show that y2 + x 3 = 0 is an implicit...Ch. 1.2 - In Problems 38, determine whether the given...Ch. 1.2 - In Problems 38, determine whether the given...Ch. 1.2 - In Problems 38, determine whether the given...Ch. 1.2 - In Problems 38, determine whether the given...Ch. 1.2 - In Problems 38, determine whether the given...Ch. 1.2 - In Problems 38, determine whether the given...Ch. 1.2 - In Problems 913, determine whether the given...Ch. 1.2 - In Problems 913, determine whether the given...Ch. 1.2 - In Problems 913, determine whether the given...Ch. 1.2 - In Problems 913, determine whether the given...Ch. 1.2 - In Problems 913, determine whether the given...Ch. 1.2 - Prob. 14ECh. 1.2 - Verify that (x) = 2/(1 cex), where c is an...Ch. 1.2 - Verify that x2 + cy2 = 1, where c is an arbitrary...Ch. 1.2 - Show that (x) = Ce3x + 1 is a solution to dy/dx ...Ch. 1.2 - Let c 0. Show that the function (x) = (c2 x2) 1...Ch. 1.2 - Prob. 19ECh. 1.2 - Determine for which values of m the function (x) =...Ch. 1.2 - Prob. 21ECh. 1.2 - Prob. 22ECh. 1.2 - Prob. 23ECh. 1.2 - In Problem 2328, determine whether Theorem 1...Ch. 1.2 - In Problem 2328, determine whether Theorem 1...Ch. 1.2 - (a) Find the total area between f(x) = x3 x and...Ch. 1.2 - In Problem 2328, determine whether Theorem 1...Ch. 1.2 - In Problem 2328, determine whether Theorem 1...Ch. 1.2 - (a) For the initial value problem (12) of Example...Ch. 1.2 - Prob. 30ECh. 1.2 - Consider the equation of Example 5, (13)ydydx4x=0....Ch. 1.3 - The direction field for dy/dx = 4x/y is shown in...Ch. 1.3 - Prob. 2ECh. 1.3 - A model for the velocity at time t of a certain...Ch. 1.3 - Prob. 4ECh. 1.3 - The logistic equation for the population (in...Ch. 1.3 - Consider the differential equation dydx=x+siny....Ch. 1.3 - Consider the differential equation dpdt=p(p1)(2p)...Ch. 1.3 - The motion of a set of particles moving along the...Ch. 1.3 - Let (x) denote the solution to the initial value...Ch. 1.3 - Use a computer software package to sketch the...Ch. 1.3 - Prob. 11ECh. 1.3 - Prob. 12ECh. 1.3 - Prob. 13ECh. 1.3 - In Problems 11-16, draw the isoclines with their...Ch. 1.3 - Prob. 15ECh. 1.3 - Prob. 16ECh. 1.3 - From a sketch of the direction field, what can one...Ch. 1.3 - Prob. 18ECh. 1.3 - Prob. 19ECh. 1.3 - Prob. 20ECh. 1.4 - In many of the problems below, it will be helpful...Ch. 1.4 - Prob. 2ECh. 1.4 - Prob. 3ECh. 1.4 - Prob. 4ECh. 1.4 - Prob. 5ECh. 1.4 - Use Eulers method with step size h = 0.2 to...Ch. 1.4 - Prob. 7ECh. 1.4 - Prob. 8ECh. 1.4 - Prob. 9ECh. 1.4 - Use the strategy of Example 3 to find a value of h...Ch. 1.4 - Prob. 11ECh. 1.4 - Prob. 12ECh. 1.4 - Prob. 13ECh. 1.4 - Prob. 14ECh. 1.4 - Prob. 15ECh. 1.4 - Prob. 16ECh. 1 - In Problems 16, identify the independent variable,...Ch. 1 - Prob. 2RPCh. 1 - Prob. 3RPCh. 1 - Prob. 4RPCh. 1 - Prob. 5RPCh. 1 - Prob. 6RPCh. 1 - Prob. 7RPCh. 1 - Prob. 8RPCh. 1 - Prob. 9RPCh. 1 - Prob. 10RPCh. 1 - Prob. 11RPCh. 1 - Prob. 12RPCh. 1 - Prob. 13RPCh. 1 - Prob. 14RPCh. 1 - Prob. 15RPCh. 1 - Prob. 16RPCh. 1 - Prob. 17RPCh. 1 - Prob. 1TWECh. 1 - Compare the different types of solutions discussed...
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- 2. Let {X} be a moving average process of order q (usually written as MA(q)) defined on tЄ Z as where {et} is a white noise process with variance 1. (1) (a) Show that for any MA(1) process with B₁ 1 there exists another MA(1) pro- cess with the same autocorrelation function, and find the lag 1 moving average coefficient (say) of this process. (b) For an MA(2) process, equation (1) becomes X=&t+B₁et-1+ B2ɛt-2- (2) i. Define the backshift operator B, and write equation (2) in terms of a polyno- mial function B(B), giving a clear definition of this function. ii. Hence show that equation (2) can be written as an infinite order autoregressive process under certain conditions on B(B), clearly stating these conditions.arrow_forwardRefer to page 92 for a problem involving solving coupled first-order ODEs using Laplace transforms. Instructions: Solve step-by-step using Laplace transforms. Show detailed algebraic manipulations and inversions. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing] Refer to page 86 for a problem involving solving Legendre's differential equation. Instructions: Solve using power series or standard solutions. Clearly justify every step and avoid unnecessary explanations. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440AZF/view?usp=sharing]arrow_forwardConsider the time series model X₁ = u(t)+s(t) + εt. Assuming the standard notation used in this module, what do each of the terms Xt, u(t), s(t) and & represent? In a plot of X against t, what features would you look for to determine whether the terms μ(t) and s(t) are required? Explain why μ(t) and s(t) are functions of t, whilst t is a subscript in X and εt.arrow_forward
- Refer to page 86 for a problem involving solving Legendre's differential equation. Instructions: Solve using power series or standard solutions. Clearly justify every step and avoid unnecessary explanations. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qo Hazb9tC440 AZF/view?usp=sharing] Refer to page 80 for a proof of convergence for a given series using the ratio test. Instructions: Clearly apply the ratio test. Show all steps and provide justification for convergence or divergence. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]arrow_forwardRefer to page 90 for a problem requiring Fourier series expansion of a given periodic function. Instructions: Clearly outline the process of finding Fourier coefficients. Provide all calculations, integrals, and final expansions. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing] Refer to page 93 for a problem involving Cauchy-Euler differential equations. Instructions: Solve the given differential equation step-by-step, showing the characteristic roots and general solution clearly.arrow_forwardRefer to page 80 for a proof of convergence for a given series using the ratio test. Instructions: Clearly apply the ratio test. Show all steps and provide justification for convergence or divergence. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440 AZF/view?usp=sharing] Refer to page 94 for a problem requiring the numerical solution of an ODE using the Runge- Kutta method. Instructions: Solve step-by-step, showing iterations, step sizes, and calculations clearly. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qo Hazb9tC440AZF/view?usp=sharing]arrow_forward
- Refer to page 82 for a double integral problem. Convert the integral into polar coordinates and evaluate it step-by-step, clearly showing all transformations and limits. Instructions: Focus only on the problem. Provide all steps, including the coordinate transformation, Jacobian factor, and the integral evaluation. Avoid irrelevant details. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 81 for a proof involving the uniqueness of solutions for a given ordinary differential equation. Instructions: Focus strictly on proving the uniqueness theorem using necessary conditions. Justify all intermediate steps. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]arrow_forwardRefer to page 88 for a problem on solving a Laplace equation in polar coordinates with boundary conditions. Instructions: Solve step-by-step using separation of variables. Clearly show transformations and solutions. Avoid irrelevant details. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440AZF/view?usp=sharing]arrow_forward
- Refer to page 89 for a line integral problem. Apply Green's Theorem to convert the line integral into a double integral. Solve it step-by-step, showing all calculations and transformations. Instructions: Outline the problem clearly. Focus on applying Green's Theorem correctly and show all double integral calculations. Avoid irrelevant content. Link [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]arrow_forwardOn page 85, a power series is given. Use the root test to prove its convergence or divergence and determine its radius of convergence. Instructions: Solve step-by-step. Apply the root test rigorously, and show all intermediate calculations. Avoid irrelevant details. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440AZF/view?usp=sharing]arrow_forwardRefer to page 77 for a problem requiring the Taylor series expansion of a function about a given point. Derive the series up to the specified order, showing all intermediate steps. Instructions: Focus on deriving the Taylor series. Clearly outline all steps, including finding derivatives and substituting into the series formula. Show detailed calculations. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZ F/view?usp=sharing]arrow_forward
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