Differential and Difference Equations(差分与微分方程)(本科教材)

Chapter 1Basic difference equations models001

1.1Difference equations of financial mathematics001

1.1.1Compound interest and loan repayments001

1.1.2Some Money Related Models002

1.2Difference equations of population theory004

1.2.1Single equations for unstructured population models004

1.2.2Structured populations and linear systems of difference equations006

1.2.3Markov chain008



Chapter 2Basic differential equations models010

2.1Equations related to financial mathematics010

2.2Continuous population models011

2.3Equations of motion: second order equations015

2.4Modelling interacting quantities systems of differential equations018



Chapter 3Solution and applications of difference equations021

3.1Linear first-order difference equations021

3.2Difference calculus and general theory of linear difference equations024

3.2.1Difference calculus025

3.2.2General theory of linear difference equations027

3.3Linear Homogeneous equations with constant coefficients033

3.4Linear Nonhomogeneous equations037

3.5Limiting behavior of solution041

3.6Autonomous（Time-Invariant）Systems043

3.7Exercises043



Chapter 4Concepts and solutions of differential equations047

4.1Concepts047

4.2Existence and uniqueness of solutions052

4.3First-order linear differential equations056

4.4Exact equation and separation of variables062

4.5Integrating factors068

4.6Initial-value and two-point boundary-value071

4.7Exercises074



Chapter 5Second and higher order differential equations077

5.1Algebraic properties of solutions077

5.2Linear equations with constant coefficients085

5.3The non-homogeneous equation092

5.4Higher order differential equations096

5.5The Euler equation103

5.6Exercises105



Chapter 6Systems of differential equations106

6.1Existence and uniqueness theorem106

6.1.1Marks and definitions106

6.1.2Existence and uniqueness of solutions112

6.2General theory of linear differential systems117

6.2.1Linear homogeneous systems117

6.2.2Linear inhomogeneous systems123

6.3Linear differential systems with constant coefficients126

6.3.1Definition and properties of matrix exponent expA126

6.3.2Calculation of fundamental solution matrix129

6.4Exercises141



Chapter 7Qualitative and stability theories147

7.1Two-dimensional autonomous system and phase plane147

7.2Plane singularity155

7.2.1Trajectory distribution of two-dimensional linear systems156

7.2.2Distribution of orbits of two-dimensional nonlinear systems in the neighborhood of singularities165

7.3Limit cycle167

7.4Lyapunov stability169

7.4.1Stability169

7.4.2First approximation theory173

7.5Exercises178



Appendix182

A.1Solution of difference equations182

A.1.1First order linear constant coefficient difference equation182

A.1.2Higher order linear constant coefficient difference equation184

A.1.3Linear constant coefficient difference equations185

A.2Solutions of ordinary differential equations186

A.2.1Symbolic solutions186

A.2.2Numerical solutions189

A.3Exercises195



References196