Elementary Differential Equations

Elementary Differential Equations integrates the underlying theory, the solution procedures, and the numerical/computational aspects of differential equations in a seamless way. For example, whenever a new type of problem is introduced (such as first-order equations, higher-order equations, systems of differential equations, etc.) the text begins with the basic existence-uniqueness theory. This provides the student the necessary framework to understand and solve differential equations. Theory is presented as simply as possible with an emphasis on how to use it. The Table of Contents is comprehensive and allows flexibility for instructors.

Table of Contents:
1. Introduction to Differential Equations. Examples of Differential Equations. Direction Fields. 2. First Order Linear Differential Equations. Existence and Uniqueness. First Order Linear Homogeneous Differential Equations. First Order Linear Nonhomogeneous Differential Equations. Introduction to Mathematical Models. Mixing Problems and Cooling Problems. 3. First Order Nonlinear Differential Equations. Existence and Uniqueness. Separable First Order Equations. Exact Differential Equations. Bernoulli Equations. The Logistic Population Model. One-Dimensional Motion with Air Resistance. One-Dimensional Dynamics with Distance as the Independent Variable. Euler's Method. 4. Second Order Linear Differential Equations. Existence and Uniqueness. The General Solution of Homogeneous Equations. Fundamental Sets and Linear Independence. Constant Coefficient Homogeneous Equations. Real Repeated Roots; Reduction of Order. Complex Roots. Unforced Mechanical Vibrations. The General Solution of the Linear Nonhomogeneous Equation. The Method of Undetermined Coefficients. The Method of Variation of Parameters. Forced Mechanical Vibrations, Electrical Networks, and Resonance. 5. Higher Order Linear Differential Equations. Existence and Uniqueness. The General Solution of nth Order Linear Homogeneous Equation. Fundamental Sets and Linear Independence. Constant Coefficient Homogeneous Equations. Nonhomogeneous Linear Equations. 6. First Order Linear Systems. The Calculus of Matrix Functions. Existence and Uniqueness. Homogeneous Linear Systems. Fundamental Sets and Linear Independence. Constant Coefficient Homogeneous Systems. Complex Eigenvalues. Repeated Eigenvalues. Nonhomogeneous Linear Systems. Euler's Method for Systems of Differential Equations. Diagonalization. Propagator Matrices, Functions of a Matrix and the Exponential Matrix. 7. Laplace Transforms. The Laplace Transform. Laplace Transform Pairs. Review of Partial Fractions. Solving Scalar Problems. Laplace Transforms of Periodic Functions. Solving Systems of Differential Equations. Convolution. The Delta Function and Impulse Response. 8. Nonlinear Systems. Existence and Uniqueness. Equilibrium Solutions and Direction Fields. Conservative Systems. Stability. Linearization and the Local Picture. The Two-dimensional Linear System y1=Ay. Predator-Prey Population Models. 9. Numerical Methods. Introduction. Euler's Method, Heun's Method, the Modified Euler's Method. Taylor Series Methods. Runge-Kutta Methods. 10. Series Solution of Differential Equations. Review of Power Series. Series Solutions near an Ordinary Point. The Euler Equation. Solutions Near a Regular Singular Point; the Method of Frobenius. The Method of Frobenius Continued; Special Cases and a Summary.