此篇文章主要在探討六個在書中常見的微分方程式，利用數學軟體Mathematica 6 繪出各個微分方程式解之幾何圖形，並從中討論各微分方程式中，係數和該微分方程式解之幾何圖形的關係，以及其對應的物理問題；最後，比較各個微分方程式有何相異或相同處。

The main topic of this article discusses six differential equations which we usually see them in the books, and we use the software of Mathematica 6 to plot their solution curves. We also discusse the relationship between coefficients and solution curves in each differential equation corresponding to its physical problem. Finally, we compare with six differential equations to find the similarities and differences.

Chapter 0. Introduction 1
Chapter 1. Oscillator 3
1.1 Simple Harmonic Oscillator 3
1.2 Damped Harmonic Oscillator 10
1.3 Theory of Oscillation 19
Chapter 2. Pendulum 29
2.1 Ideal Pendulum 32
2.2 The Nonlinear Pendulum with Friction 36
2.3 Theory of Pendulum 41
Chapter 3. Van Der Pol Equation 46
Chapter 4. Duffing Equation 59
4.1 Unforced Duffing Equation 60
4.2 Forced Duffing Equation 68
Chapter 5. Predator-Prey Equation(Volterra-Lotka Equation) 77
5.1 Predator-Prey Models with Constant Growth 78
5.2 Predator-Prey Models with Limited Growth 84
5.3 Introduce Volterra and Lotka 87
Chapter 6. Lorenz Equation 90
Chapter 7. Comparison and Conclusion 99
Appendix Mathematica codes 101
References 130

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