Chapter 1 Basic difference equations models 1.1 Difference equations of financial mathematics 1.1.1 Compound interest and loan repayments 1.1.2 Some Money Related Models 1.2 Difference equations of population theory 1.2.1 Single equations for unstructured population models 1.2.2 Structured populations and linear systems of difference equations 1.2.3 Markov chain Chapter 2 Basic differential equations models 2.1 Equations related to financial mathematics 2.2 Continuous population models 2.3 Equations of motion: second order equations 2.4 Modelling interacting quantities systems of differential equations Chapter 3 Solution and applications of difference equations 3.1 Linear first-order difference equations 3.2 Difference calculus and general theory of linear difference equations 3.2.1 Difference calculus 3.2.2 General theory of linear difference equations 3.3 Linear Homogeneous equations with constant coefficients 3.4 Linear Nonhomogeneous equations 3.5 Limiting behavior of solution 3.6 Autonomous（Time-Invariant）Systems 3.7 Exercises Chapter 4 Concepts and solutions of differential equations 4.1 Concepts 4.2 Existence and uniqueness of solutions 4.3 First-order linear differential equations 4.4 Exact equation and separation of variables 4.5 Integrating factors 4.6 Initial-value and two-point boundary-value 4.7 Exercises Chapter 5 Second and higher order differential equations 5.1 Algebraic properties of solutions 5.2 Linear equations with constant coefficients 5.3 The non-homogeneous equation 5.4 Higher order differential equations 5.5 The Euler equation 5.6 Exercises Chapter 6 Systems of differential equations 6.1 Existence and uniqueness theorem 6.1.1 Marks and definitions 6.1.2 Existence and uniqueness of solutions 6.2 General theory of linear differential systems 6.2.1 Linear homogeneous systems 6.2.2 Linear inhomogeneous systems 6.3 Linear differential systems with constant coefficients 6.3.1 Definition and properties of matrix exponent expA 6.3.2 Calculation of fundamental solution matrix 6.4 Exercises Chapter 7 Qualitative and stability theories 7.1 Two-dimensional autonomous system and phase plane 7.2 Plane singularity 7.2.1 Trajectory distribution of two-dimensional linear systems 7.2.2 Distribution of orbits of two-dimensional nonlinear systems in the neighborhood of singularities 7.3 Limit cycle 7.4 Lyapunov stability 7.4.1 Stability 7.4.2 First approximation theory 7.5 Exercises Appendix A.1 Solution of difference equations A.1.1 First order linear constant coefficient difference equation A.1.2 Higher order linear constant coefficient difference equation A.1.3 Linear constant coefficient difference equations A.2 Solutions of ordinary differential equations A.2.1 Symbolic solutions A.2.2 Numerical solutions A.3 Exercises References
