一阶微分方程图解法教学要点分析

Teaching key analysis of phase graphic solution method of first-order differential equations

通过分析一阶微分方程解曲线切线斜率取值的特点,找出了解曲线的几何意义.根据解曲线的几何意义,以一个具体的一阶方程为例讲解了画出一阶方程近似解的3步法,直观地展示了一阶微分方程的几何意义.利用计算机数值模拟了具体的方程,得到了2类特殊的一阶微分方程线素场的特点以及在通解中常数C的几何意义.

Through analyzing the value characteristics of the slope of of first order differential equations, find the geometric significance of solution curves. According to the geometric significance of solution curves, by taking a specific first order differential equation as an example, illustrated the three-step method of sketching the solution graphs, and intuitionally show the geometry significance of the first order differential equation. Finally, by using numerical simulation, obtain the slope field characteristics of two special first-order differential equations, and the geometry significance of the constant C in the general solution.