2-4 Exact Equations 2-4-1 方法的條件 任何 first order DE 皆可改寫成 的型態 (1) 當 成立時， (2) 當 is independent of x 或 is independent of y 可以用Modified Exact Equation Method 來解 (見講義 2-4-5)

2-4-2 方法的來源 Review the concept of partial differentiation Specially, when f(x, y) = c where c is some constant,

補充： 思考：假設一個人在山坡的某處。若往東走，每走 1 公尺，高度會增加 a 公尺。若往北走，每走 1 公尺，高度會增加 b 公尺。假設這人現在所在的位置是(0, 0)。那麼這人的東北方，座標為 (p, q) 的地方，高度會比 (0, 0) 高多少？ a  p + b  q (p, q) p q (p, 0) (0, 0) a b

[Definition 2.4.1] We can express any 1st order DE as  If there exists some function f(x, y) that satisfies and , then we call the 1st order DE the exact equation.  The method for checking whether the DE is an exact equation: (Proof): If and , then

For the exact equation, 可改寫成

2-4-3 解法 The method for solving the exact equation (A): (2) (1) (2) g(y) is a constant for x (2) (3) 代入 (3) (2) (4) further computation Solution

整理 Previous Step: Check whether is satisfied. Step 1: Solve Step 2: 將 f(x, y) 代入 ，以解出 g(y) Step 3: Substitute g(y) into Step 4: Further computation and obtain the solution Extra Steps: (a) Consider the initial value problem

The method for solving the exact equation (B): (2) (2) (1) (2) h(x) is a constant for y (3) 代入 (3) (2) (4) further computation Solution

2-4-4 例子 Example 1 (text page 65) Step 0: check whether it is exact

Example 2 (text page 65) Step 0: check for exact Step 1 Step 2 Step 3 Step 3 Step 2 Step 4

要注意 (a) 自行由另一個方向 來練習， 看是否得出同樣的結果。 (b) 得出的解 為 implicit solution (c) 思考：何時用 何時用

Example 3 (text page 66) 自修，但注意 initial value problem, 何時用 (c) 得出的 implicit solution 為 , 範圍：x  (-1, 1) 而 explicit solution 為 , 範圍： x  (-1, 1) 為何 不為解？

2-4-5 Modified Exact Equation Method Technique: Use the integrating factor (x, y) to convert the 1st order DE into the exact equation. such that It is hard to find .

(1) When is a function of y alone: We can set  to be dependent on y alone. Therefore, 用 separable variable 的方法

(2) When is a function of x alone: We can set  to be dependent on x alone. Therefore, 用 separable variable 的方法

前面 2-4-3 (pages 105~107) 的解法流程再加一個步驟： Step 0: Case 1 Yes (小心易背錯) Use the process of 2-4-3 Case 2 Whether Yes Whether Whether No is independent of x No is independent of y Case 3 In Cases 2 and 3, Yes Case 4 No Use other methods Using the process of 2-4-3, but M(x, y) should be modified as M(x, y) N(x, y) should be modified as N(x, y)

(independent of x) (Case 2) Example 4 (text page 67) Step 0: (independent of x) (Case 2) Q: 為何 c 以及  可省略? double N Steps 1~4:

2-4-6 本節需要注意的地方 (1) 使本節方法時，要先將 DE 改成如下的型態 並且假設 2-4-6 本節需要注意的地方 (1) 使本節方法時，要先將 DE 改成如下的型態 並且假設 (2) 對 x 而言，g(y) 是個常數；對 y 而言，h(x) 是個常數 (3) 本節很少有 singular solution 的問題， 但是可能有 singular point 的問題 (4) 背熟三個判別式，二種情況的 integrating factor (小心勿背錯) 並熟悉解法的流程

2-5 Solutions by Substitutions 介紹 3 個特殊解法 Question: 尚有不少的 1st order DE 無法用 Sections 2-2~2-4 的方法來解 本節所提到的特殊解法的共通點： 用新的變數 u 來取代 y 對 症 下 藥

2-5-1 特殊解法 1: Homogeneous Equations If g(tx, ty) = tαg(x, y), then g(x, y) is a homogeneous function of degree α. Which one is homogeneous? g(x, y) = x3 + y3 g(x, y) = x3 + y3 +1 注意：課本中，homogeneous 有兩種定義 一種是 Section 2-3 的定義 (較常用) 一種是這裡的定義 兩者並不相同

If M(x, y) and N(x, y) are homogeneous functions of the same degree,  For a 1st order DE: If M(x, y) and N(x, y) are homogeneous functions of the same degree, then the 1st order DE is homogeneous. It can by solved by setting y = xu, dy = udx + xdu, and use the separable value method. 解法的限制條件 (from page 102)

If is homogeneous then where u = y/x, y = xu 以 t = 1/x 得出 (separable)

Procedure for solving the homogeneous 1st order DE Previous Step: Conclude whether the DE is homogeneous (快速判斷法：看 powers (指數) 之和) Step 1: Set y = ux, dy = udx + xdu 並化簡 Step 2: Convert into the separable 1st order DE Step 3: Solve the separable 1st order DE (用 Sec. 2-2 的方法) Step 4: Substitute u = y/x (別忘了這個步驟)

Example 1 (text page 71) M(x, y) N(x, y) Previous Step: Conclude whether the DE is homogeneous M(tx, ty) = t2M(x, y) N(tx, ty) = t2N(x, y) homogeneous DE Step 1: Set y = ux, dy = udx + xdu 原式 Step 2: Convert into the separable 1st order DE

Step 3: Solve the separable 1st order DE Step 4 代回 u = y/x

2-5-2 特殊解法 2: Bernoulli’s Equations We can set u = y1–n , , and the method of solving the 1st order linear DE to solve the Bernoulli’s equation. so (Chain rule)

Procedure for solving the Bernoulli’s equation Previous Step : Conclude whether the DE is a Bernoulli’s equation Step 1: Set , Step 2: Convert the Bernoulli’s equation into the 1st order linear DE Step 3: Solve the 1st order linear DE (use the method in Sec. 2-3) Step 4: Substitute u = y1–n (別忘了)

Example 2 (text page 77) Previous Step: 判斷 (Bernoulli, n = 2) Step 1: set u = y–1 (y = u–1) (Chain rule) Step 2: Convert into the 1st order linear DE (standard form) 原式 Step 3: Obtain the solution of the 1st order DE Step 4: 代回 u = y–1

2-5-3 特殊解法 3 If the 1st order DE has the form, (B  0) 2-5-3 特殊解法 3 If the 1st order DE has the form, we can set u = Ax + By + C to solve it. (B  0) (解法的限制條件) Since (這式子也許較好記)

Procedure for solving Previous Step: Conclude Step 1: Set Step 2: Converting (轉化成用其他方法可以解出來的 DE 未必一定是轉化成 separable variable DE) Step 3: Solving Step 4: Substitute (別忘了)

Example 3 (text page 77) , Previous Step: 判斷 Step 1: Set Step 2: Converting 原式 Step 3: Obtain the solution (別忘了在運算過程中，代回 u = Ax + By) 注意 的算法 Step 4: 代回 u = Ax + By +C

2-5-4 本節要注意的地方 (1) 對症下藥，先判斷 DE 符合什麼樣的條件，再決定要什麼方法來解 (部分的 DE 可以用兩個以上的方法來解) (2) 別忘了，寫出解答時，要將 u 用 y/x, y1-n, 或 Ax + By + C 代回來 (3) 本節方法皆有五大步驟 Previous Step: 判斷用什麼方法 Step 1: Set u = …, du/dx = … Step 2: Converting， Step 3: Solving， Step 4: 將 u 用 x, y 代回來

整理：Methods of solving the 1st order DE Direct computation (2) Separable variable (3) Linear DE (4) Exact equation 破解法： 直接積分 條件： 破解法： x, y 各歸一邊後積分 條件： 破解法：算 條件： 破解法：double N method 先處理 再處理 條件： (或反過來)

(4-1) Exact equation 變型 (4-2) Exact equation 變型 (5) Homogeneous equation 破解法： 條件： is exact independent of x 破解法： 條件： is exact independent of y 破解法：u = y/x, (y = xu) 再用 separable variable method 條件：

(6) Bernoulli’s Equation (7) Ax + By + C 破解法： u = y1–n 再用 linear DE 的方法 條件： 破解法： u = Ax + By + c 條件： 注意 (a) 速度的訓練 (b) Exercises in Review 2多練習 (c) 行有餘力，觀察 singular solution 和 singular point

練習題 Sec. 2-4: 3, 8, 13, 17, 25, 29, 32, 34, 35, 38, 42 Sec. 2-5: 3, 5, 10, 13, 14, 17, 20, 22, 24, 25, 29, 31 Chap. 2 Review: 2, 13 , 16, 17, 18, 19, 22, 23, 24, 26, 27