Student Resource Manual to accompany Differential Equations: A Modeling Perspective, 2e

by Borrelli, Robert L.; Coleman, Courtney S.

• ISBN13: 9780471433330
• ISBN10: 0471433330
• Edition: 2nd
• Format: Paperback
• Copyright: 2004-04-10
• Publisher: Wiley

Summary

Work more effectively and gauge your progress along the way! This Student Resource Manual contains worked-out solutions to approximately half of the problems in Borrelli's Differential Equations, 2 nd Edition. In addition to problem solutions, it offers graphs, suggestions for students and additional resource material.With the modeling and graphical visualization as the central approach, Borrelli's Differential Equations, 2 nd Edition introduces differential systems and numerical methods early on and encourages the use of numerical solvers from the very start. It covers modern topics such as sensitivity, long-term behavior, bifurcation, and chaos together with the basic solution formula techniques and theory.

Author Biography

Robert L. Borrelli is the author of Student Resource Manual to accompany Differential Equations: A Modeling Perspective, 2e, published by Wiley. Courtney S. Coleman is the author of Student Resource Manual to accompany Differential Equations: A Modeling Perspective, 2e, published by Wiley.

Table of Contents

1 Modeling and Differential Equations	1	(29)
1.1 The Modeling Approach	1	(4)
1.2 A Modeling Adventure	5	(3)
1.3 Models and Initial Value Problems	8	(7)
1.4 The Modeling Process: Differential Systems	15	(15)
Spotlight on Modeling: Radiocarbon Dating	19	(6)
Spotlight on Modeling: Cold Medication I	25	(5)
2 First Order Differential Equations
2.1 Linear Differential Equations	30	(3)
2.2 Linear Differential Equations: Qualitative Analysis	33	(5)
2.3 Existence and Uniqueness of Solutions	38	(5)
2.4 Visualizing Solution Curves: Slope Fields	43	(7)
2.5 Separable Differential Equations: Planar Systems	50	(5)
2.6 A Predator-Prey Model: the Lotka-Volterra System	55	(4)
2.7 Extension of Solutions: Long-Term Behavior	59	(5)
2.8 Qualitative Analysis: State Lines, Sign Analysis	64	(5)
2.9 Bifurcations: A Harvested Logistic Model	69	(29)
Spotlight on Approximate Numerical Solutions	73	(3)
Spotlight on Computer Implementation	76	(6)
Spotlight on Steady States: Linear ODEs	82	(4)
Spotlight on Modeling: Cold Medication II	86	(1)
Spotlight on Change of Variables: Pursuit Models	87	(9)
Spotlight on Continuity in the Data	96	(2)
3 Second Order Differential Equations	98	(52)
3.1 Models of Springs	99	(5)
3.2 Undriven Constant-Coefficient Linear Differential Equations	104	(4)
3.3 Visualizing Graphs of Solutions: Direction Fields	108	(5)
3.4 Periodic Solutions: Simple Harmonic Motion	113	(3)
3.5 Driven Linear ODEs: Undetermined Coefficients I	116	(3)
3.6 Driven Linear ODEs: Undetermined Coefficients II	119	(8)
3.7 Theory of Second-Order Linear Differential Equations	127	(7)
3.8 Nonlinear Second-Order Differential Equations	134	(16)
Spotlight on Modeling: Vertical Motion	142	(5)
Spotlight on Modeling: Shock Absorbers	147	(1)
Spotlight on Einstein's Field Equations	147	(3)
4 Applications of Second Order Differential Equations	150	(16)
4.1 Newton's Laws: The Pendulum	150	(5)
4.2 Beats and Resonance	155	(3)
4.3 Frequency Response Modeling	158	(4)
4.4 Electrical Circuits	162	(4)
Spotlight on Modeling: Tuning a Circuit	165	(1)
5 The Laplace Transform	166	(25)
5.1 The Laplace Transform: Solving IVPs	167	(4)
5.2 Working with the Transform	171	(4)
5.3 Transforms of Periodic Functions	175	(5)
5.4 Convolution	180	(11)
Spotlight on the Delta Function	184	(4)
Spotlight on Modeling: Time Delays and Collisions	188	(3)
6 Linear Systems of Differential Equations	191	(53)
6.1 Compartment Models: Tracking Lead	192	(7)
6.2 Eigenvalues, Eigenvectors, and Eigenspaces of Matrices	199	(3)
6.3 Undriven Linear Differential Systems: Real Eigenvalues	202	(6)
6.4 Undriven Linear Systems: Complex Eigenvalues	208	(7)
6.5 Orbital Portraits for Planar Systems	215	(5)
6.6 Driven Systems: The Matrix Exponential	220	(9)
6.7 Steady States	229	(4)
6.8 The Theory of General Linear Systems	233	(11)
Spotlight on Vectors, Matrices, Independence	236	(4)
Spotlight on Linear Algebraic Equations	240	(2)
Spotlight on Bifurcations: Sensitivity	242	(2)
7 Nonlinear Differential Systems	244	(18)
7.1 Chemical Kinetics: The Fundamental Theorem	245	(4)
7.2 Properties of Autonomous Systems, Direction Fields	249	(5)
7.3 Interacting Species: Cooperation, Competition	254	(8)
Spotlight on Modeling: Destructive Competition	260	(1)
Spotlight on Modeling: Bifurcation and Sensitivity	261	(1)
8 Stability	262	(23)
8.1 Stability and Linear Autonomous Systems	263	(4)
8.2 Stability and Nonlinear Autonomous Systems	267	(8)
8.3 Conservative Systems	275	(10)
Spotlight on Lyapunov Functions	279	(5)
Spotlight on Rotating Bodies	284	(1)
9 Nonlinear Systems: Cycles and Chaos	285	(28)
9.1 Cycles	286	(9)
9.2 Solution Behavior in Planar Autonomous Systems	295	(5)
9.3 Bifurcations	300	(7)
9.4 Chaos	307	(7)
Spotlight on Chaotic Systems	309	(4)
10 Fourier Series and Partial Differential Equations	313	(37)
10.1 Vibrations of a Guitar String	314	(7)
10.2 Fourier Trigonometric Series	321	(5)
10.3 Half-Range and Exponential Fourier Series	326	(3)
10.4 Temperature in a Thin Rod	329	(4)
10.5 Sturm-Liouville Problems	333	(4)
10.6 The Method of Eigenfunction Expansions	337	(14)
Spotlight on Decay Estimates	342	(1)
Spotlight on the Optimal Depth for a Wine Cellar	342	(1)
Spotlight on Approximation of Functions	343	(7)
11 Series Solutions	350	(21)
11.1 The Method of Power Series	351	(4)
11.2 Series Solutions near an Ordinary Point	355	(4)
11.3 Regular Singular Points: The Euler ODE	359	(2)
11.4 Series Solutions near Regular Singular Points	361	(10)
Spotlight on Legendre Polynomials	365	(3)
Spotlight on Bessel Functions	368	(3)
A. Basic Theory of Initial Value Problems	371
A.1 Uniqueness	371	(1)
A.2 The Picard Process for Solving an Initial Value Problem	372	(3)
A.3 Extension of Solutions	375