Program Explanation Given information: The system of differential equations is shown below. { x ′ 1 = 9 x 1 + 2 x 2 x ′ 2 = 2 x 1 + 6 x 2 Explanation: The system of differential equations can be written in x ′ = A x form as, x ′ = [ 9 2 2 6 ] x Where the coefficient matrix A is described as, A = [ 9 2 2 6 ] So, obtain the characteristic equation of matrix A to obtain the eigenvalues as, | 9 − λ 2 2 6 − λ | = 0 ( 9 − λ ) ( 6 − λ ) − 4 = 0 λ 2 − 15 λ + 50 = 0 Further, solve the above equation and obtain the eigenvalues as, λ 1 = 10 or λ 2 = 5 Now, obtain the eigenvector corresponding to λ 1 = 10 as follows: [ 9 − 10 2 2 6 − 10 ] [ a b ] = [ 0 0 ] [ − 1 2 2 − 4 ] [ a b ] = [ 0 0 ] [ − a + 2 b 2 a − 4 b ] = [ 0 0 ] From the comparison of corresponding elements from both sides and simplification, it is obtained that a = 2 b . For simplest vector, consider 2 for a and Therefore, b = 1 , and the eigenvector corresponding to λ 1 = 10 is written as, v 1 = [ 2 1 ] Similarly, obtain the eigenvector corresponding to λ 2 = 5 as follows: [ 9 − 5 2 2 6 − 5 ] [ a b ] = [ 0 0 ] [ 4 2 2 1 ] [ a b ] = [ 0 0 ] [ 4 a + 2 b 2 a + b ] = [ 0 0 ] From the comparison of corresponding elements from both sides and simplification, it is obtained that 2 a = − b

5th Edition

ISBN: 9780135988510

Author: Edwards

Publisher: PEARSON CO

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Problems on Growing Functions

For a given set of functions f and g, the growth of functions states that the function g grows as fast as the function f. The growth rate of algorithms helps determine the algorithm's efficiency and compare its performance to other algorithms.

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Chapter 5.6, Problem 19P

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Program Description: Purpose of the problem is to obtain value of an exponential matrix eAt for the system x′=Ax .

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Chapter 5.6, Problem 18P

Chapter 5.6, Problem 20P

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The correct answer is D Could you explain and show the steps please

Taylor Series Approximation Example- H.W More terms used implies better approximation f(x) 4 f(x) Zero order f(x + 1) = f(x;) First order f(x; + 1) = f(x;) + f'(x;)h 1.0 Second order 0.5 True f(x + 1) = f(x) + f'(x)h + ƒ"(x;) h2 2! f(x+1) 0 x; = 0 x+1 = 1 x h f(x)=0.1x4-0.15x³- 0.5x2 -0.25x + 1.2 51 Taylor Series Approximation H.w: Smaller step size implies smaller error Errors f(x) + f(x,) Zero order f(x,+ 1) = f(x) First order 1.0 0.5 Reduced step size Second order True f(x + 1) = f(x) + f'(x)h f(x; + 1) = f(x) + f'(x)h + "(xi) h2 f(x,+1) O x₁ = 0 x+1=1 Using Taylor Series Expansion estimate f(1.35) with x0 =0.75 with 5 iterations (or & s= 5%) for f(x)=0.1x 0.15x³-0.5x²- 0.25x + 1.2 52

Could you explain this using the formula I attached and polar coorindates

Chapter 5 Solutions

EP DIFF.EQUAT.+BOUND.VALUE,...UPD.-ACC.

Ch. 5.1 - Let A=[2347] and B=[3451]. Find (a) 2A+3B; (b)...Ch. 5.1 - Prob. 2P Ch. 5.1 - Find AB and BA given A=[203415] and B=[137032].Ch. 5.1 - Prob. 4P Ch. 5.1 - Prob. 5P Ch. 5.1 - Prob. 6P Ch. 5.1 - Prob. 7P Ch. 5.1 - Prob. 8P Ch. 5.1 - Prob. 9P Ch. 5.1 - Prob. 10P

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Program Description: Purpose of the problem is to obtain value of an exponential matrix e A t for the system x ′ = A x .

Solution Summary: The author explains that the problem is to obtain value of an exponential matrix eAt for the system of differential equations.

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Program Description: Purpose of the problem is to obtain value of an exponential matrix eAt for the system x′=Ax .

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Chapter 5 Solutions