5507. 【清華冬令營2018模擬】取石子
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AP Calculus - 考前學的微分方程
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<time title="建立時間:2020-04-18 16:40:59 / 修改時間:20:41:58" itemprop="dateCreated datePublished" datetime="2020-04-18T16:40:59+08:00">2020-04-18</time>
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<h1 id="緒論"><a href="#緒論" class="headerlink" title="緒論"></a>緒論</h1><p>本來因該在去年暑假就把所有東西全部自學完的,但是那個時候太忙太懶,於是乎心理安慰自己學完了。但是,馬上考試了,不學就卒,遂花了兩小時學。</p>
微分方程
有時我們需要通過未知函式及其導數所滿足的關係式去求未知函式,這種關係是就是微分方程。例如:
但上面這個方程在這裡不做討論,因為不考。
通解、特解
微分方程的解有兩種形式,一種解是含有任意常數且任意常數的個數正好與方程的階數相同,這樣的解成為微分方程的通解(General Solution)。另一種解中不含任意常數,稱為特解(Specific Solution),通常可按照問題所給的條件從通解中確定任意常數的待定值來求特解。
注:在考慮通解中,並不需要考慮其是否處處可微(比如一些隱函式,圓的方程),但是特解需要考慮。例如:通解可以是 但特解要寫成:(其中取)解微分方程
在這裡,我們只討論形如
或
其中皆在考慮的範圍內連續。這種形式的方程,我們稱其為 可分離變數的一劫微分方程,因為其可以轉換為
的形式。
求解的方法即為對兩邊分別積分:
斜率場 - 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.
斜率場(slope field)是座標系當中一組小的線段,代表的是一個微分方程(differential equation)的函式解(solutions)的影象(solution curves)的影象的切線的斜率。
作者:commontolerance
連結:https://www.zhihu.com/question/53824255/answer/1029489348
來源:知乎
著作權歸作者所有。商業轉載請聯絡作者獲得授權,非商業轉載請註明出處。
解題方法總結:
- 關注某些特殊點,例如
- 關注,之間的關係
- 觀察,是否獨立,比方說影象是否可以直接水平平移或者豎直平移
- 觀察變化趨勢
尤拉方法 - Euler’s Method
總結:使用點斜率和一個Initial Point來估算原函式。例如一個函式,其導數為。已知在其影象上,那麼以為步長,可以估算出原函式的幾個點:
指數增長和衰減 - Exponential Growth & Decay
滿足
的函式,就符合指數型地增長或衰減。其中,當時為Exponential Growth,時為Exponential Decay。
之所以稱其為指數型增長,是因為原函式
特點:
有限增長 - Restricted Growth
滿足
的函式,符合Restricted Growth,其原函式為:
特點:
對數增長 - Logistic Growth
滿足
的函式符合Logistic Growth。原函式為:
特點:
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