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      AP Calculus - 考前学的微分方程
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          <time title="创建时间:2020-04-18 16:40:59 / 修改时间:20:41:58" itemprop="dateCreated datePublished" datetime="2020-04-18T16:40:59+08:00">2020-04-18</time>
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    <h1 id="绪论"><a href="#绪论" class="headerlink" title="绪论"></a>绪论</h1><p>本来因该在去年暑假就把所有东西全部自学完的,但是那个时候太忙太懒,于是乎心理安慰自己学完了。但是,马上考试了,不学就卒,遂花了两小时学。</p>

微分方程

有时我们需要通过未知函数及其导数所满足的关系式去求未知函数,这种关系是就是微分方程。例如:

但上面这个方程在这里不做讨论,因为不考。

通解、特解

微分方程的解有两种形式,一种解是含有任意常数且任意常数的个数正好与方程的阶数相同,这样的解成为微分方程的通解(General Solution)。另一种解中不含任意常数,称为特解(Specific Solution),通常可按照问题所给的条件从通解中确定任意常数的待定值来求特解。

注:在考虑通解中,并不需要考虑其是否处处可微(比如一些隐函数,圆的方程),但是特解需要考虑。例如:通解可以是 但特解要写成:(其中取)

解微分方程

在这里,我们只讨论形如

其中皆在考虑的范围内连续。这种形式的方程,我们称其为 可分离变量的一劫微分方程,因为其可以转换为

的形式。

求解的方法即为对两边分别积分:

斜率场 - Slope Fields

Slope field is a set of little line segments in the coordinate system representing the slope of tangent lines for the solution curves of a given differential equation. Moreover, it is a graphical tool intended to find out the solution to the differential equations, generally at first order.

斜率场(slope field)是坐标系当中一组小的线段,代表的是一个微分方程(differential equation)的函数解(solutions)的图像(solution curves)的图像的切线的斜率。

作者:commontolerance
链接:https://www.zhihu.com/question/53824255/answer/1029489348
来源:知乎
著作权归作者所有。商业转载请联系作者获得授权,非商业转载请注明出处。

解题方法总结:

  • 关注某些特殊点,例如
  • 关注,之间的关系
  • 观察,是否独立,比方说图像是否可以直接水平平移或者竖直平移
  • 观察变化趋势

欧拉方法 - Euler’s Method

总结:使用点斜率和一个Initial Point来估算原函数。例如一个函数,其导数为。已知在其图像上,那么以为步长,可以估算出原函数的几个点:

指数增长和衰减 - Exponential Growth & Decay

满足

的函数,就符合指数型地增长或衰减。其中,当时为Exponential Growth,时为Exponential Decay。

之所以称其为指数型增长,是因为原函数

特点:

有限增长 - Restricted Growth

满足

的函数,符合Restricted Growth,其原函数为:

特点:

对数增长 - Logistic Growth

满足

的函数符合Logistic Growth。原函数为:

特点:

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本文标题:AP Calculus - 考前学的微分方程

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发布时间:2020年04月18日 - 16:04

最后更新:2020年04月18日 - 20:04

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