Differential equations for engineers: the essentials

Content: Machine generated contents note: ch. 1 Introduction --
1.1. Foundations --
1.1.1. Arithmetic --
1.1.2. Algebra --
1.1.2.1. Quadratic Equations --
1.1.2.2. Higher-Order Polynomial Equations --
1.1.2.3. Systems of Simultaneous Linear Equations --
1.1.2.4. Logarithms and Exponentials --
1.1.3. Trigonometry --
1.1.4. Differential Calculus --
1.1.4.1. Derivative --
1.1.4.2. Rules of Differentiation --
1.1.4.3. Infinite Series --
1.1.5. Integral Calculus --
1.1.5.1. Change of Variable --
1.1.5.2. Integration by Parts --
1.1.6. Transitioning to First-Order Ordinary Differential Equations --
1.2. Classes of Differential Equations --
1.3. A Few Other Things --
1.4. Summary --
Solved Problems --
Problems to Solve --
Word Problems --
Challenge Problems --
ch. 2 First-Order Linear Ordinary Differential Equations --
2.1. First-Order Linear Homogeneous Ordinary Differential Equations --
2.1.1. Example: RC Circuit --
2.1.2. Time-Varying Equations and Justification of Separation of Variables Procedure --
2.2. First-Order Linear Nonhomogeneous Ordinary Differential Equations --
2.2.1. Integrating Factor --
2.2.2. Kernel and Convolution --
2.2.3. System Viewpoint --
2.3. System Stability --
2.4. Summary --
Solved Problems --
Problems to Solve --
Word Problems --
Challenge Problems --
ch. 3 First-Order Nonlinear Ordinary Differential Equations --
3.1. Nonlinear Separable Ordinary Differential Equations --
3.1.1. Example: Sounding Rocket Trajectory --
3.1.1.1. Maximum Speed Going Up --
3.1.1.2. Maximum Altitude --
3.1.1.3. Maximum Speed Coming Down --
3.2. Possibility of Transforming Nonlinear Equations into Linear Ones --
3.3. Successive Approximations for almost Linear Systems --
3.4. Summary --
Solved Problems --
Problems to Solve --
Word Problems --
Challenge Problems --
Computing Projects: Phase 1 --
ch. 4 Existence, Uniqueness, and Qualitative Analysis --
4.1. Motivation --
4.2. Successive Approximations (General Form) --
4.2.1. Successive Approximations Example --
4.3. Existence and Uniqueness Theorem --
4.3.1. Lipschitz Condition --
4.3.2. Statement of the Existence and Uniqueness Theorem --
4.3.3. Successive Approximations Converge to a Solution --
4.4. Qualitative Analysis --
4.5. Stability Revisited --
4.6. Local Solutions --
4.7. Summary --
Solved Problems --
Problems to Solve --
Word Problems --
Challenge Problems --
ch. 5 Second-Order Linear Ordinary Differential Equations --
5.1. Second-Order Linear Time-Invariant Homogeneous Ordinary Differential Equations --
5.1.1. Two Real Roots --
5.1.2. Repeated Roots --
5.1.3. Complex Conjugate Roots --
5.1.4. Under-, Over-, and Critically Damped Systems --
5.1.5. Stability of Second-Order Systems --
5.2. Fundamental Solutions --
5.3. Second-Order Linear Nonhomogeneous Ordinary Differential Equations --
5.3.1. Example: Automobile Cruise Control --
5.3.2. Variation of Parameters (Kernel) Method --
5.3.3. Undetermined Coefficients Method --
5.3.3.1. Steady-State Solutions --
5.3.3.2. Example: RLC Circuit with Sine-Wave Source --
5.3.3.3. Resonance with a Sine-Wave Source --
5.3.3.4. Resonance with a Periodic Input Other than a Sine Wave --
5.3.3.5. Satisfying Initial Conditions Using the Undetermined Coefficients Method --
5.3.3.6. Limitations of the Undetermined Coefficients Method --
5.4. Existence and Uniqueness of Solutions to Second-Order Linear Ordinary Differential Equations --
5.5. Summary --
Solved Problems --
Problems to Solve --
Word Problems --
Challenge Problems --
ch. 6 Higher-Order Linear Ordinary Differential Equations --
6.1. Example: Satellite Orbit Decay --
6.2. Higher-Order Homogeneous Linear Equations --
6.3. Higher-Order Nonhomogeneous Linear Equations --
6.3.1. Variation of Parameters (Kernel) Method --
6.3.2. Undetermined Coefficients Method --
6.4. Existence and Uniqueness --
6.5. Stability of Higher-Order Systems --
6.6. Summary --
Solved Problems --
Problems to Solve --
Word Problems --
Challenge Problems --
Computing Projects: Phase 2 --
ch. 7 Laplace Transforms --
7.1. Preliminaries --
7.2. Introducing the Laplace Transform --
7.3. Laplace Transforms of Some Common Functions --
7.4. Laplace Transforms of Derivatives --
7.5. Laplace Transforms in Homogeneous Ordinary Differential Equations --
7.6. Laplace Transforms in Nonhomogeneous Ordinary Differential Equations --
7.6.1. General Solution Using Laplace Transforms --
7.6.2. Laplace Transform of a Convolution --
7.6.3. Differential Equations with Impulsive and Discontinuous Inputs --
7.6.3.1. Defining the Dirac Delta Function --
7.6.3.2. Ordinary Differential Equations with Impulsive Inputs --
7.6.3.3. Ordinary Differential Equations with Discontinuous Inputs --
7.6.4. Initial and Final Value Theorems --
7.6.5. Ordinary Differential Equations with Periodic Inputs --
7.6.5.1. Response to a Periodic Input --
7.6.5.2. Resonance with a Nonsinusoidal Periodic Input --
7.7. Summary --
Solved Problems --
Problems to Solve --
Word Problems --
Challenge Problems --
ch. 8 Systems of First-Order Ordinary Differential Equations --
8.1. Preliminaries --
8.2. Existence and Uniqueness --
8.3. Numerical Integration Methods --
8.3.1. Euler Method --
8.3.2. Runge --
Kutta Method --
8.3.3. Comparing Accuracy of Euler and Runge --
Kutta Methods --
8.3.4. Variable Step Size --
8.3.5. Other Numerical Integration Algorithms --
8.4. Review of Matrix Algebra --
8.5. Linear Systems in State Space Format --
8.5.1. Homogeneous Linear Time-Invariant Systems --
8.5.1.1. Real, Distinct Eigenvalues: Heat Transfer Example --
8.5.1.2. Complex Eigenvalues: Electric Circuit Example --
8.5.1.3. Complex Eigenvalues: Aircraft Dynamics Example --
8.5.1.4. Repeated Eigenvalues --
8.5.1.5. State Transition Matrix --
8.5.1.6. Linear System Stability in State Space --
8.5.1.7. Coordinate Systems: Road Vehicle Dynamics --
8.5.1.8. Coordinate Systems: A General Analytic Approach --
8.5.1.9. A System Viewpoint on Coordinate Systems --
8.5.2. Nonhomogeneous Linear Systems --
8.5.2.1. Matrix Kernel Method --
8.5.2.2. Matrix Laplace Transform Method --
8.5.2.3. Matrix Undetermined Coefficients Method --
8.5.2.4. Worth of the Three Methods, in Summary --
8.6. A Brief Look at N-Dimensional Nonlinear Systems --
8.6.1. Equilibrium Points --
8.6.2. Stability --
8.6.3. Imposition of Limits --
8.6.4. Uniquely Nonlinear Dynamics --
8.6.5. Mathematical Modeling Tools for Nonlinear Systems --
8.7. Summary --
Solved Problems --
Problems to Solve --
Word Problems --
Challenge Problems --
ch. 9 Partial Differential Equations --
9.1. Heat Equation --
9.1.1. Heat Equation in Initial Value Problems --
9.1.1.1. Heat Equation in an Example Initial Value Problem --
9.1.1.2. Existence and Uniqueness of Solutions to Parabolic Partial Differential Equations in Initial Value Problems --
9.1.2. Heat Equation in Boundary Value Problems --
9.2. Wave Equation and Power Series Solutions --
9.2.1. Wave Equation in Initial Value Problems --
9.2.1.1. Traveling Plane Wave --
9.2.1.2. Traveling Wave in Spherical Coordinates --
9.2.2. Wave Equation in Boundary Value Problems --
9.2.2.1. Derivation of the Wave Equation for a Taut Membrane --
9.2.2.2. Solving the Wave Equation in a Boundary Value Problem in Cylindrical Coordinates --
9.2.3. Power Series Solutions: An Introduction --
9.2.4. Bessel Equations and Bessel Functions --
9.2.4.1. Bessel Functions of the First Kind --
9.2.4.2. Bessel Functions of the Second Kind --
9.2.5. Power Series Solutions: Another Example --
9.2.6. Power Series Solutions: Some General Rules --
9.2.6.1. A Few Definitions --
9.2.6.2. Some General Rules --
9.2.6.3. Special Cases --
9.3. Laplace's Equation --
9.3.1. Laplace's Equation Example in Spherical Coordinates --
9.4. Beam Equation --
9.4.1. Deriving the Beam Equation --
9.4.2. Beam Equation in a Boundary Value Problem --
9.4.3. Orthogonality of the Beam Equation Eigenfunctions --
9.4.4. Nonhomogenous Boundary Value Problems --
9.4.5. Other Geometries --
9.5. Summary --
Solved Problems --
Problems to Solve --
Word Problems --
Challenge Problems.