Complex Variables and Applications (9th Edition) by Brown & Churchill – Mathematics Solutions Manual

All Chapters Covered




SOLUTION MANUAL

, Table of Contents

Part I Ordinary Differential Equations
1 Introduction to Differential Equations 1

2 First-Order Differential Equations 22

3 Higher-Order Differential Equations 99

4 The Laplace Transform 198

5 Series Solutions of Linear Differential Equations 252

6 Numerical Solutions of Ordinary Differential Equations 317

Part II Vectors, Matrices, and Vector Calculus
7 Vectors 339

8 Matrices 373

9 Vector Calculus 438

Part III Systems of Differential Equations
10 Systems of Linear Differential Equations 551

11 Systems of Nonlinear Differential Equations 604

Part IV Fourier Series and Partial Differential Equations
12 Orthogonal Functions and Fourier Series 634

13 Boundary-Value Problems in Rectangular Coordinates 680

14 Boundary-Value Problems in Other Coordinate Systems 755

15 Integral Transform Method 793

16 Numerical Solutions of Partial Differential Equations 832

, Part V Complex Analysis
17 Functions of a Complex Variable 854

18 Integration in the Complex Plane 877

19 Series and Residues 896

20 Conformal Mappings 919

Appendices
Appendix II Gamma function 942

Projects
3.7 Road Mirages 944

3.10 The Ballistic Pendulum 946

8.1 Two-Ports in Electrical Circuits 947

8.2 Traffic Flow 948

8.15 Temperature Dependence of Resistivity 949

9.16 Minimal Surfaces 950

14.3 The Hydrogen Atom 952

15.4 The Uncertainity Inequality in Signal Processing 955

15.4 Fraunhofer Diffraction by a Circular Aperture 958

16.2 Instabilities of Numerical Methods 960

, Part I Ordinary Differential Equations


Introduction to
1 Differential Equations

EXERCISES 1.1
Definitions and Terminology


1. Second order; linear ph ph



2. Third order; nonlinear because of (dy/dx)4
ph ph ph ph ph



3. Fourth order; linear ph ph



4. Second order; nonlinear because of cos(r + u)
ph ph ph ph ph ph ph

!
5. Second order; nonlinear because of (dy/dx)2 or
ph ph ph ph ph ph 1+ p h (dy/dx)2
6. Second order; nonlinear because of R2
ph ph ph ph ph



7. Third order; linear ph ph



8. Second order; nonlinear because of x˙2
ph ph ph ph ph



9. Writing the differential equation in the form x(dy/dx) + y2 = 1, we see that it is nonlinear in y
ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph


phbecause phof phy . ph However, phwriting phit ph in ph the phform ph (y ph− ph1)(dx/dy) ph+ phx ph= ph0, ph we phsee phthat ph it phis phlinear
2 2

ph in x.ph



10. Writing the differential equation in the form u(dv/du) + (1 + u)v = ueu we see that it is linear in
ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph


v. However, writing it in the form (v + uv − ueu)(du/dv) + u = 0, we see that it is nonlinear
ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph


in u.
ph ph



11. From y = e−x/2 we obtain y′ 2 = − 1 e−x/2. Then 2y′ + y = −e−x/2 + e−x/2 = 0.
ph ph ph ph ph ph ph ph ph
ph
ph ph ph ph ph ph ph ph ph ph


12. From y = 6 − 6 e−20t we obtain dy/dt = 24e−20t, so that
ph ph ph ph ph
ph
ph ph ph ph ph ph ph
5 5
" # ph
dy 6 6 −20t
+ 20y = 24e −20t + 20 − e = 24.
dt 5 5
13. From y = e3x cos 2x ph p h p h ph ph p h we p h obtain p h y′ p h = p h 3e3x cos 2x − 2e3x sin 2x
ph ph ph ph ph ph p h and p h y′′ p h = p h 5e3x cos 2x −
ph ph ph


12e3x sin 2x, so that
ph ph ph p h p h


y′′ − 6y′ + 13y = 0.
ph ph ph ph ph ph



14. From y = − cos x ln(sec x + tan x) we obtain y′ = −1 + sin x ln(sec x + tan x) and
ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph


y′′ = tan x + cos x ln(sec x + tan x). Then y′′ + y = tan x.
ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph



15. The domain of the function, found by solving x + 2 ≥ 0, is [−2, ∞). From y′ = 1 + 2(x + 2)−1/2 we
ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph ph

have
ph

—
(y − x)y′ = (y − x)[1 + (2(x + 1/2
ph 2)] ph ph ph ph




1