# Differential Equations (DE)

## Presentation on theme: "Differential Equations (DE)"— Presentation transcript:

Differential Equations (DE)

9/12 2. 9/19 3. 9/26 4. 10/3 5. 10/10 國慶日 6. 10/17 7. 10/24 8. 10/31 9. 11/7: Midterms 範圍： (Sections 2-2 ~ 4-5) 10. 11/14 11. 11/21 12. 11/28 13. 12/5 14. 12/12 15. 12/19  12/19 小考 16. 12/26 17. 1/2 18. 1/9: Finals  範圍： (Sections 4-6 ~ 12-1)

Higher Order DE 應用 (Sec. 5-1) 非線性 (Sec. 4-10, Sec. 5-3, 微方2) 多項式解法 (Chap. 6，微方2) 解法 (Sec. 12-1) Partial DE 直角座標 (Chapter 12，微方2) 圓座標 (Chapter 13，微方2) Laplace Transform (Chap. 7) Transforms Fourier Series (Chap. 11) Fourier Transform (Chap. 14，微方2)

Chapter 1 Introduction to Differential Equations
1.1 Definitions and Terminology (術語) Differential Equation (DE): any equation containing derivation (text page 3, definition 1.1) x: independent variable 自變數 y(x): dependent variable 應變數

Note: In the text book, f(x) is often simplified as f
notations of differentiation , , , , ………. Leibniz notation , , , , ………. prime notation , , , , ………. dot notation , , , , ………. subscript notation

(2) Ordinary Differential Equation (ODE):
differentiation with respect to one independent variable (3) Partial Differential Equation (PDE): differentiation with respect to two or more independent variables

(4) Order of a Differentiation Equation: the order of the highest derivative in the equation
7th order 2nd order

(5) Linear Differentiation Equation:
All of the coefficient terms am(x) m = 1, 2, …, n are independent of y. Property of linear differentiation equations: If and y3 = by1 + cy2, then (if g(x) is treated as the input and y(x) is the output)

(6) Non-Linear Differentiation Equation

(7) Explicit Solution (text page 8)
The solution is expressed as y = (x) (8) Implicit Solution (text page 8) Example: , Solution: (implicit solution) or (explicit solution)

1.2 Initial Value Problem (IVP)
A differentiation equation always has more than one solution. for , y = x, y = x+1 , y = x+2 … are all the solutions of the above differentiation equation. General form of the solution: y = x+ c, where c is any constant. The initial value (未必在 x = 0) is helpful for obtain the unique solution. and y(0) = y = x+2 and y(2) = y = x+1.5

The kth order differential equation usually requires k initial conditions (or k boundary conditions) to obtain the unique solution. solution: y = x2/2 + bx + c, b and c can be any constant y(1) = 2 and y(2) = 3 y(0) = 1 and y'(0) =5 y(0) = 1 and y'(3) =2 For the kth order differential equation, the initial conditions can be 0th ~ (k–1)th derivatives at some points. (boundary conditions，在不同點) (initial conditions ，在相同點) (boundary conditions，在不同點)

1.3 Differential Equations as Mathematical Model
Physical meaning of differentiation: the variation at certain time or certain place Example 1: x(t): location, v(t): velocity, a(t): acceleration F: force, β: coefficient of friction, m: mass

Example 2: 人口隨著時間而增加的模型
A: population 人口增加量和人口呈正比

Example 3: 開水溫度隨著時間會變冷的模型
T: 熱開水溫度, Tm: 環境溫度 t: 時間

DE Review dependent variable and independent variable PDE and ODE
Order of DE linear DE and nonlinear DE explicit solution and implicit solution initial value; boundary value IVP

Chapter 2 First Order Differential Equation
2-1 Solution Curves without a Solution Instead of using analytic methods, the DE can be solved by graphs (圖解) slopes and the field directions: y-axis the slope is f(x0, y0) (x0, y0) x-axis

Example dy/dx = 0.2xy From： Fig (a) in “Differential Equations-with Boundary-Value Problem”, 8th ed., Dennis G. Zill and Michael R. Cullen.

Example 2 dy/dx = sin(y), y(0) = –3/2
With initial conditions, one curve can be obtained From： Fig in “Differential Equations-with Boundary-Value Problem”, 8th ed., Dennis G. Zill and Michael R. Cullen.

Advantage: It can solve some 1st order DEs that cannot be solved by mathematics. Disadvantage: It can only be used for the case of the 1st order DE. It requires a lot of time

Section 2-6 A Numerical Method
Another way to solve the DE without analytic methods independent variable x x0, x1, x2, ………… Find the solution of Since approximation sampling(取樣) 前一點的值 取樣間格

Example: dy(x)/dx = 0.2xy y(xn+1) = y(xn) + 0.2xn y(xn )*(xn+1 –xn). dy/dx = sin(x) y(xn+1) = y(xn) + sin(xn)*(xn+1 –xn). 後頁為 dy/dx = sin(x), y(0) = –1, (a) xn+1 –xn = 0.01, (b) xn+1 –xn = 0.1, (c) xn+1 –xn = 1, (d) xn+1 –xn = 0.1, dy/dx = 10sin(10x) 的例子 Constraint for obtaining accurate results: (1) small sampling interval (2) small variation of f(x, y)

(a) (b) (c) (d)

Advantages -- It can solve some 1st order DEs that cannot be solved by mathematics. -- can be used for solving a complicated DE (not constrained for the 1st order case) -- suitable for computer simulation Disadvantages -- numerical error (數值方法的課程對此有詳細探討)

Exercises for Practicing
(not homework, but are encouraged to practice) 1-1: 1, 13, 19, 23, 37 1-2: 3, 13, 21, 33 1-3: 2, 7, 28 2-1: 1, 13, 20, 25, 33 2-6: 1, 3