Differential equations for engineers: the essentials

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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.