Fundamentals of Differential Equations

(Most chapters end with a Chapter Summary, Review Problems and Group Projects.)

1. Introduction.


Background.



Solutions and Initial Value Problems.



Direction Fields.



The Approximation Method of Euler.



2. First Order Differential Equations.


Introduction: Motion of a Falling Body.



Separable Equations.



Linear Equations.



Exact Equations.



Special Integrating Factors.



Substitutions and Transformations.



3. Mathematical Models and Numerical Methods Involving First Order Equations.


Mathematical Modeling.



Compartmental Analysis.



Heating and Cooling of Buildings.



Newtonian Mechanics.



Electrical Circuits.



Improved Euler's Method.



Higher-Order Numerical Methods: Taylor and Runge-Kutta.



4. Linear Second Order Equations.


Introduction: The Mass-Spring Oscillator.



Homogeneous Linear Equations; the General Solution.



Auxiliary Equations with Complex Roots.



Nonhomogeneous Equations: the Method of Undetermined Coefficients.



The Superposition Principle and Undetermined Coefficients Revisited.



Variation of Parameters.



Qualitative Considerations for Variable-Coefficient and Nonlinear Equations.



A Closer Look at Free Mechanical Vibrations.



A Closer Look at Forced Mechanical Vibrations.



5. Introduction to Systems and Phase Plane Analysis.


Interconnected Fluid Tanks.



Elimination Method for Systems with Constant Coefficients.



Solving Systems and Higher-Order Equations Numerically.



Introduction to the Phase Plane.



Coupled Mass-Spring Systems.



Electrical Systems.



Dynamical Systems, Poincaré Maps, and Chaos.



6. Theory of Higher-Order Linear Differential Equations.


Basic Theory of Linear Differential Equations.



Homogeneous Linear Equations with Constant Coefficients.



Undetermined Coefficients and the Annihilator Method.



Method of Variation of Parameters.



7. Laplace Transforms.


Introduction: A Mixing Problem.



Definition of the Laplace Transform.



Properties of the Laplace Transform.



Inverse Laplace Transform.



Solving Initial Value Problems.



Transforms of Discontinuous and Periodic Functions.



Convolution.



Impulses and the Dirac Delta Function.



Solving Linear Systems with Laplace Transforms.



8. Series Solutions of Differential Equations.


Introduction: The Taylor Polynomial Approximation.



Power Series and Analytic Functions.



Power Series Solutions to Linear Differential Equations.



Equations with Analytic Coefficients.



Cauchy-Euler (Equidimensional) Equations.



Method of Frobenius.



Finding a Second Linearly Independent Solution.



Special Functions.



9. Matrix Methods for Linear Systems.


Introduction.



Review 1: Linear Algebraic Equations.



Review 2: Matrices and Vectors.



Linear Systems in Normal Form.



Homogeneous Linear Systems with Constant Coefficients.



Complex Eigenvalues.



Nonhomogeneous Linear Systems.



The Matrix Exponential Function.



10. Partial Differential Equations.


Introduction: A Model for Heat Flow.



Method of Separation of Variables.



Fourier Series.



Fourier Cosine and Sine Series.



The Heat Equation.



The Wave Equation.



Laplace's Equation.



Appendices.


Newton's Method.



Simpson's Rule.



Cramer's Rule.



Method of Least Squares.



Runge-Kutta Precedure for n Equations.



Answers to Odd-Numbered Problems.


Index.