Chapter 4 Higher Order Differential Equations.ppt

* Example 1 (text page 130) We have known that y1 = ex is one of the solution P(x) = 0 Specially, set c = –2, (y2(x) 只要 independent of y1(x) 即可 所以 c 的值可以任意設) General solution: 4.2.3 例子 * We have known that y1 = x2 is one of the solution Note: the interval of x If x ? (0, ?) (x 0), 如課本 If x 0, Example 2 (text page 131) when x ? (??, 0) (將課本 x 的範圍做更改) * 4.2.4 本節需注意的地方 (1) 記住公式 (2) 若不背公式 (不建議)，在計算過程中別忘了對 w(x) 做積分 (3) 別忘了 P(x) 是 “standard form” 一次微分項的 coefficient term (4) 同樣有 singular point 的問題 (5) 因為 y2(x) 是 homogeneous linear DE 的 “任意解”，所以計算時，常數的選擇以方便為原則 (6) 由於 的計算較複雜且花時間，所以要多加練習 多算習題 * 附錄六： Hyperbolic Function 比較： * * * * 附錄七 Linear DE 解法的步驟 (參照講義 page 151) Step 1: Find the general solution (i.e., the complementary function ) of the associated homogeneous DE Step 2: Find the particular solution Step 3: Combine the complementary function and the particular solution (Sections 4-2, 4-3, 4-7) (Sections 4-4, 4-5, 4-6) Extra Step: Consider the initial (or boundary) conditions * 4-3 Homogeneous Linear Equations with Constant Coefficients 4-3-1 限制條件 本節使用 auxiliary equation 的方法來解 homogeneous DE 限制條件: (1) homogeneous (2) linear (3) constant coefficients KK: [ ? ? ? ? ? ? ] (the simplest case of the higher order DEs) a0, a1, a2, …. , an are constants * Suppose that the solutions has the form of emx Example: y(x) ?3 y(x) + 2 y(x) = 0 Set y(x) = emx, m2 emx ? 3m emx + 2 emx = 0 m2 ? 3m + 2 = 0 solve m 4-3-2 解法 可以直接把 n 次微分用 mn 取代，變成一個多項式 這個多項式被稱為 auxiliary equation 解法核心： * ? 解法流程 Step 1-1 auxiliary function Find n roots , m1, m2, m3, …., mn (If m1, m2, m3, …., mn are distinct) n linearly independent solutions Complementaryfunction Step 1-1 Step 1-2 Step 1-3 (有三個 Cases) * roots Case 1 m1 ? m2, m1, m2 are real solutions 4-3-3 Three C