Chapter 1 Firstorder Differential Equation 1 1 Ordinary

程數學 Chapter 1 First-order Differential Equation 楊學成老師

單元架構 1 -1 Ordinary differential equations and their applications 1 -2 Geometrical Meaning of

1. 1 First-order Differential Equation: Ordinary differential equations and their applications 標學習目學習完本小節後，您將能：

1. 3 Separable Differential Equations 標學習目學習完本小節後，您將能： 1. 能說出什麼是Separable D. E. 2. 能具體描述Separable

1. 5 Linear Differential Equations Bernoulli Eq. 標學習目學習完本小節後，您將能： 1. 能說出 Linear Differential

Basic concepts ‧ Voltage drop : potential difference.

1. 7 Orthogonal Trajectories of cwiuto. 標學習目學習完本小節後，您將能： 1. 能說出什麼是Orthogonal 2. 能具體描述Equipotential line

Basic concepts • Orthogonal perpendiculan • Goal: Using differential eq for finding cwuto ineerseto

How to find orthogonal Trajectories • 1 st step: • • the given circuits

1. 8 Existence and Uniqueness of Solutions Picard Iteration. 標學習目學習完本小節後，您將能： 1. 能說出何謂Uniqueness公式

Basic concepts • At least one solution • At most one solution existence: theorems,

Basic concepts Then Theorem 2: Uniqueness theorem

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程數學 Chapter 1 First-order Differential Equation 楊學成老師

單元架構 1 -1 Ordinary differential equations and their applications 1 -2 Geometrical Meaning of Direction Fields 1 -3 Separable Differential Equations 1 -4 Exact differential Equation 1 -5 Linear Differential Equations Bernoulli Eq. 1 -6 Modeling: Electric circuits. 1 -7 Orthogonal Trajectories of cwiuto. 1 -8 Existence and Uniqueness of Solutions Picard Iteration.

1. 1 First-order Differential Equation: Ordinary differential equations and their applications 標學習目學習完本小節後，您將能： 1. 能說出什麼是ODE 2. 能寫出ODE的方程式 3. 能計算基本的ODE方程式 4. 能說出ODE可應用的地方

Basic concepts

Basic concepts

1. 3 Separable Differential Equations 標學習目學習完本小節後，您將能： 1. 能說出什麼是Separable D. E. 2. 能具體描述Separable D. E. 的特性 3. 能寫出Separable的公式 4. 能進行Separable D. E. 的運算

Basic concepts

Modeling : Separable Equation

Basic concepts

1. 5 Linear Differential Equations Bernoulli Eq. 標學習目學習完本小節後，您將能： 1. 能說出 Linear Differential Equations 2. 能具體描述Linear Differential Equations 3. 能寫出Linear Differential Equations 4. 能進行Linear Differential Equations 計算

Basic concepts

Basic concepts ‧ Voltage drop : potential difference.

Kirchhoff's Voltage Law

Basic concepts

1. 7 Orthogonal Trajectories of cwiuto. 標學習目學習完本小節後，您將能： 1. 能說出什麼是Orthogonal 2. 能具體描述Equipotential line 3. 能寫出Orthogonal Trajectories的特性 4. 能找出Orthogonal Trajectories

Basic concepts • Orthogonal perpendiculan • Goal: Using differential eq for finding cwuto ineerseto given cwuto at right angle. • 同位能 • (equipotential line)

How to find orthogonal Trajectories • 1 st step: • • the given circuits are solution curves • 2 nd step: • orthogonal trajectories. • ------(1) • 3 rd step: • solve the differential eq(1) • The solution of eq(1) can write as • ------(2) • The eq (2) is called a one – parameters family of curves.

Basic concepts

1. 8 Existence and Uniqueness of Solutions Picard Iteration. 標學習目學習完本小節後，您將能： 1. 能說出何謂Uniqueness公式 2. 能具體描述Solutions的存在條件 3. 能寫出計算Existence solutions的過程 4. 能進行具有initial value的運算

Basic concepts • At least one solution • At most one solution existence: theorems, uniqueness Theorem 1: Existence theorem If f(x, y) is continues at all points (x, y) in some rectangle (Fig, 31) R: , and bounded in R, say for all(x, y) in R.

Basic concepts Then Theorem 2: Uniqueness theorem

Basic concepts

Basic concepts • Initial Value Problem:

Basic concepts