Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 1 First-Order Differential Equations A differential equation defines a relationship between an unknown function and one or more of its derivatives Applicable to: –Chemistry –Physics –Engineering –Medicine –Biology

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung 微分方程式的分類 Ordinary Differential Equation : (ODE) y 只與一個變數 x 有關 Partial Differential Equation : (PDE) u 與兩個變數 x, y 有關 Total Differential Equation : (TDE) Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung First order differential equation with y as the dependent variable and x as the independent variable would be: Second order differential equation would have the form: 常微分方程式 An ordinary differential equation is one with a single independent variable. The order ( 階 ) of an equation: The order of the highest derivative appearing in the equation Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Linear – if the nth-order differential equation can be written: a n (t)y (n) + a n-1 (t)y (n-1) a 1 y’ + a 0 (t)y = h(t) Nonlinear – not linear Example of an Ordinary Differential Equation If f(x) = 0, The ODE is Homogeneous If f(x)  0, The ODE is Non-homogeneous 最高階導數的次數 (degree) 稱為此微分方程式之次數 Example : Degree = 1Degree = 2 Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 1 First-Order Differential Equations 初值問題與邊界問題

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung General Solution – all solutions to the differential equation can be represented in this form for all constants Particular Solution – contains no arbitrary constants   )( )( sin cos solutionparticular definediscIf solutiongeneral cxy xy   Chapter 1 First-Order Differential Equations 通解與特殊解

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung 奇異解 (Singular Solutions) A differential equation may sometimes have an additional solution that cannot be obtained from the general solution. solutionaalsois x yparabolabut ccxy solutiongeneralwith yyxy    Chapter 1 First-Order Differential Equations 顯式解 (explicit solution)

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung 可分離微分方程式 (Separable Differential Equations) Special form 分離變數 (Separable Variables) Example: Boyle’s gas law Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Example 1 : Solve the differential equation Chapter 1 First-Order Differential Equations 原式 : 兩邊作積分

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung 試求 : 之通解 (83 清大化工 ) 先利用座標平移來消去常數 2 與 –6 令 令 Chapter 1 First-Order Differential Equations 隱式解 (implicit solution)

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 1 First-Order Differential Equations Reduction to Separable Form 1. Differential equations of the form --- 有時稱為齊次 (homogeneous) 方程式 其中 g 為 y/x 的任意函數, 例如 :, 欲求其解可令 and

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 1 First-Order Differential Equations Reduction to Separable Form 2. Transformations Example : 令 代入並簡化 乘以 2 並分離變數 積分可得 隱式解 (implicit solution)

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung First-Order Ordinary Differential Equations For φ(x,y)=C If We call Exact ( 正合 ) Differential Equation Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung 試求 : 之解 (84 中央光電 ) Exact Differential Equation Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Inexact Differential Equation Exact Differential Equation If But if Inexact Differential Equation An integrating factor Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung 解 Inexact Differential Equation However, we see that A function of x alone Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung First-Order Ordinary Differential Equations 亦可應用 全微分觀念 求通解 Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung First-Order Ordinary Differential Equations 亦可應用 全微分觀念 Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung 求通解 Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Linear First-Order Ordinary Differential Equations An integrating factor We must require Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 1 First-Order Differential Equations 試解線性微分方程式

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung RL Circuit For special case : V(t) =V 0 If the initial condition : I(0) = 0  C = -V 0 /R Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung 解 Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung 解 Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung First-Order Ordinary Differential Equations (Nonlinear  Linear) Nonlinear Linear Chapter 1 First-Order Differential Equations (Bernoulli equation)

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung 解 Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung 試解稱為 Verhuls 方程式的特殊柏努力方程式 (A,B 為正值常數 ) Chapter 1 First-Order Differential Equations (y = 0 也是一解 ) ( 稱為人口成長的 logistic law)

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 1 First-Order Differential Equations ( 稱為人口成長的 logistic law) 當 B = 0 時 指數成長模型 (Malthus’s law) 為一 “ 抑止項 ” (braking term)

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 1 First-Order Differential Equations q(x)  輸入 (input), 例如 : 力 y(x)  輸出 (output) 或是響應 (response), 例如 : 位移, 電流 …. 等 與初值相關 總輸出 = 對應於輸入的輸出 + 對應於初始數據的輸出

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Clairaut Differential Equations 通解 奇解 (singular solution) Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung solutionaalsois x yparabolabut ccxy solutiongeneralwith yyxy    Singular Solution Envelope curve ( 包絡線 ) Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung 求通解與奇解 其中 a, b 均為常數 通解 奇解 (singular solution) Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung 一般情形下, 若已知有包絡線存在, 則對於曲線 F(x,y,c)=0 求出其包絡線 的方法是求出以下之聯立解 求通解與奇解 通解 奇解 (singular solution) Chapter 1 First-Order Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 Differential Equation First-Order ODE (Second-Order  First-Order) 1. F(x, y’, y’’) = 0, 意即不含因變數 y 取 p(x) = y’  F(x, y’, y’’) = F(x, p, p’) 2. F(y, y’, y’’) = 0, 意即不含自變數 x 取 p(x) = y’  F(y, y’, y’’) = F(y, p, pp’) 3. y’’+p(x)y’+q(x)y = 0 二階線性齊次常微分方程式 取 u(x) = y’/y  y’ = yu, y’’ = y’u + yu’ ( y’u + yu’) + pyu + qy = 0  u’ +[y’/y + p]u + q = 0 一階非線性常微分方程式 Riccati equation

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 Differential Equation First-Order ODE --- Riccati equation setting v(x) is assumed to be a solution of the Riccati equation Linear First-Order Ordinary Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 Differential Equation First-Order ODE --- Picard 疊代近似法 對於初始值問題,, 若 可積分 則其解為 If …… 利用 Picard 疊代近似法計算到