Solutions Manual for Advanced Modern Engineering Mathematics 5th Ed – Burley (2019)

Covers All 12 Chapters




SOLUTIONS MANUAL

, TABLE OF CONTENTS



Page

Chapter 1. Matrix Analysis 1
Chapter 2. Numerical Solution of Ordinary Differential Equations 86
Chapter 3. Vector Calculus 126
Chapter 4. Functions of a Complex Variable 194
Chapter 5. Laplace Transforms 270
Chapter 6. The z Transform 369
Chapter 7. Fourier Series 413
Chapter 8. The Fourier Transform 489
Chapter 9. Partial Differential Equations 512
Chapter 10. Optimization 573
Chapter 11. Applied Probability and Statistics 639




iii

, 1
Matrix Analysis

Exercises 1.3.3

1(a) Yes, as the three vectors are linearly independent and span three-
dimensional space.


1(b) No, since they are linearly dependent
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
3 1 1
⎣ 2 ⎦ − 2⎣ 0⎦ = ⎣ 2 ⎦
5 1 3


1(c) No, do not span three-dimensional space. Note, they are also linearly
dependent.


2 Transformation matrix is
⎡ ⎤⎡ ⎤ ⎡ √1 ⎤
= 1 11 −1 1 0
0 1
0 0
1 00 = √ 2
√1
− √22 0
0
A √ 2⎣ ⎦⎣ ⎦ ⎣ 12 1 ⎦
√
0 0 2 0 0 1 0 0 1

Rotates the (e1, e2) plane through π/4 radians about the e3 axis.


3 By checking axioms (a)–(h) on p. 10 it is readily shown that all cubics
ax3 + bx2 + cx + d form a vector space. Note that the space is four dimensional.
3(a) All cubics can be written in the form

ax3 + bx2 + cx + d

and {1, x, x2, x3} are a linearly independent set spanning four-dimensional space.
Thus, it is an appropriate basis.



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c Pearson Education Limited 2011

, 2 Glyn James, Advanced Modern Engineering Mathematics, 4th Edition


3(b) No, does not span the required four-dimensional space. Thus a general
cubic cannot be written as a linear combination of

(1 − x), (1 + x), (1 − x3), (1 + x3)


as no term in x2 is present.



3(c) Yes as linearly independent set spanning the four-dimensional space

a(1 − x)+ b(1 + x) + c(x2 − x3) + d(x2 + x3)


= (a + b) + (b − a)x + (c + a)x2 + (d − c)x3

≡ α + βx + γx2 + δx3


3(d) Yes as a linear independent set spanning the four-dimensional space

a(x − x2) + b(x + x2) + c(1 − x3) + d(1 + x3)


= (a + b) + (b − a)x + (c + d)x2 + (d − c)x3

≡ α + βx + γx2 + δx3


3(e) No not linearly independent set as

(4x3 + 1) = (3x2 + 4x3) − (3x2 + 2x) + (1 + 2x)



4 x + 2x3, 2x − 3x5, x + x3 form a linearly independent set and form a basis
for all polynomials of the form α + βx3 + γx5 . Thus, S is the space of all odd
quadratic polynomials. It has dimension 3.



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c Pearson Education Limited 2011