I’m working on a engineering project and need guidance to help me study.

I need help with my engineering math project. It covers questions from Chapters 1-8 and varies in questions. I need all work to be written out, and to be correct. The Chapter topics are listed below so please review using the textbook link so that you know what you are doing.

THERE ARE 5-10 QUESTIONS

Chapter 1: Introduction to Differential Equations 3

1.1 Definitions and Terminology 4

1.2 Initial-Value Problems 14

1.3 Differential Equations as Mathematical Models 19

Chapter 2: First-Order Differential Equations 33

2.1 Solution Curves Without a Solution 34

2.1.1 Direction Fields 34

2.1.2 Autonomous First-Order DEs 36

2.2 Separable Equations 43

2.3 Linear Equations 50

2.4 Exact Equations 59

2.5 Solutions by Substitutions 65

2.6 A Numerical Method 69

2.7 Linear Models 74

2.8 Nonlinear Models 84

2.9 Modeling with Systems of First-Order DEs 93

Chapter 3: Higher-Order Differential Equations

3.1 Theory of Linear Equations 106

3.1.1 Initial-Value and Boundary-Value Problems 106

3.1.2 Homogeneous Equations 108

3.1.3 Nonhomogeneous Equations 113

3.2 Reduction of Order 117

3.3 Homogeneous Linear Equations with Constant Coefficients 120

3.4 Undetermined Coefficients 127

3.5 Variation of Parameters 136

3.6 Cauchy–Euler Equations 141

3.7 Nonlinear Equations 147

3.8 Linear Models: Initial-Value Problems 151

3.8.1 Spring/Mass Systems: Free Undamped Motion

3.8.2 Spring/Mass Systems: Free Damped Motion 155

3.8.3 Spring/Mass Systems: Driven Motion 158

3.8.4 Series Circuit Analogue 161

3.9 Linear Models: Boundary-Value Problems 167

3.10 Green’s Functions 177

3.10.1 Initial-Value Problems 177

3.10.2 Boundary-Value Problems 183

3.11 Nonlinear Models 187

3.12 Solving Systems of Linear Equations 196 Chapter 3 in Review 203

Chapter 4: The Laplace Transform

4.1 Definition of the Laplace Transform 212

4.2 The Inverse Transform and Transforms of Derivatives

4.2.1 Inverse Transforms 218

4.2.2 Transforms of Derivatives 220

4.3 Translation Theorems 226

4.3.1 Translation on the s-axis 226

4.3.2 Translation on the t-axis 229

4.4 Additional Operational Properties 236

4.4.1 Derivatives of Transforms 237

4.4.2 Transforms of Integrals 238

4.4.3 Transform of a Periodic Function 244

4.5 The Dirac Delta Function 248

4.6 Systems of Linear Differential Equations 251

Chapter 5: Series Solutions of Linear Differential Equations

5.1 Solutions about Ordinary Points 262

5.1.1 Review of Power Series 262

5.1.2 Power Series Solutions 264

5.2 Solutions about Singular Points 271

5.3 Special Functions 280

5.3.1 Bessel Functions 280

5.3.2 Legendre Functions 288

Chapter 5 in Review 294

Chapter 6: Numerical Solutions of Ordinary Differential Equations

6.1 Euler Methods and Error Analysis 298

6.2 Runge–Kutta Methods 302

6.3 Multistep Methods 307

6.4 Higher-Order Equations and Systems 309

6.5 Second-Order Boundary-Value Problems 313

Chapter 7: Vectors

7.1 Vectors in 2-Space 322

7.2 Vectors in 3-Space 327

7.3 Dot Product 332

7.4 Cross Product 338

7.5 Lines and Planes in 3-Space 345

7.6 Vector Spaces 351

7.7 Gram–Schmidt Orthogonalization Process 359

Chapter 8: Matrices

8.1 Matrix Algebra 368

8.2 Systems of Linear Algebraic Equations 376

8.3 Rank of a Matrix 389

8.4 Determinants 393

8.5 Properties of Determinants 399

8.6 Inverse of a Matrix 405

8.6.1 Finding the Inverse 405

8.6.2 Using the Inverse to Solve Systems 411

8.7 Cramer’s Rule 415

8.8 The Eigenvalue Problem 418

8.9 Powers of Matrices 426

8.10 Orthogonal Matrices

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