# 楊學成 老師 Chapter 1 First-order Differential Equation.

## Presentation on theme: "楊學成 老師 Chapter 1 First-order Differential Equation."— Presentation transcript:

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.

‧ Voltage drop : potential difference.

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

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.

At least one solution At most one solution existence: theorems, uniqueness Theorem1: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.

Then Theorem2: Uniqueness theorem

Initial Value Problem: