“（十）自控数学篇：常微分方程”中提到了微分方程，经知友提醒，发现了一些错误。我查阅学习后，已将前篇文章已更正，这里把相关的内容补充一下。

资料主要来源于下面这篇电子书：

Classification of differential equations​users.aber.ac.uk/ruw/teach/260/classification.php#:~:text=An%20ordinary%20differential%20equation%20(ODE,respect%20to%20several%20independent%20variables

以及维基百科：

首先，微分方程定义：

“In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.”

简单来说，工程中的微分方程就是描述物理量以及物理量变化速率的等式，其中微分对应于变化速率。根据方程形式的不同，一般可分为两类，常微分方程（ordinary differential equation，ODE）和偏微分方程partial differential equations （PDE)。它们的区别在于变量的数量，其中ODE仅含有一个独立变量。即：

“An ordinary differential equation (ODE) contains differentials with respect to only one variable, partial differential equations (PDE) contain differentials with respect to several independent variables.”

经典控制理论遇到的都是ODE。简单系统及其对应的ODE可以求解得到解析解，比如前面提到的弹簧阻尼系统，解的形式也就是指数挂上三角函数；复杂的方程就需要迭代求数值解了。

对于微分方程，可加的定语还有：“方程的阶数”；“齐次/非其次”；“线性/非线性”。这里就不展开了。感兴趣的可以继续看下面几篇知乎文章，写的很赞，传送门如下：