<commit_before>---
published: true
title: ECON351 FINAL 2017
category: econ351
tags:
  - econ351
  - Differential equations
---

重要的事情说三遍。

**Practice exam, Practice exam, Practice exam**

**刷题，刷题，刷题**

[What do top students do differently? | <NAME> |  
科学研究发现，
](https://www.youtube.com/watch?v=Na8m4GPqA30)

1. not IQ
2. not work hard
3. do Practice exam

少年，你离学霸的距离，就是少刷了一道题。

## 计划
1. 首先，把近些年 last final 做好。
2. 然后，复习近些年的midterm。
3. 最后，根据做错的题目，阅读slides。
4. 考试的时候，永远从最简单的题目，做起。不会的，难得，就留到后面。交卷前，一定检查。
-------------------------------------------------


## Homogenous and particular solution

练习例子
$$ \ddot y + a_1 \dot y + a_2 y = b$$



## 16.[Integration - Chapter 16 ](http://web.uvic.ca/~rondeau/econ351/protected/ch16slides.pdf)

### 16.1 2015 final q5

画图，帮助自己理解。


## 16.2 2016 MT2  Q1

Basic integration problem。 反求导。


[Integrate[10000 - 10 Sqrt[q] - q/10, {q, 0, 16}]](http://www.wolframalpha.com/input/?i=integrate(+10000+-+10*q%5E0.5+-+q%2F10,+0,+16))

[Plot[10000 - 10 Sqrt[q] - q/10, {q, 0, 16}]](https://sandbox.open.wolframcloud.com/app/objects/7972818e-79ce-4296-8b16-e17cd7295746#sidebar=compute)



----------------------------

## 18. [Slides Chapter 18 - First Order Autonomous Linear Difference Equations ](http://web.uvic.ca/~rondeau/econ351/protected/ch18slides.pdf)

the difference equation
 - solve
    - 不要背公式
    - 一步一步写到5步
    - 总结规律，写出close form
    - 转换成 $a^t(y_0 - \bar y)  + \bar y$
 - steady state. $\bar y$
 - the stability properties of the variable and steady state
   - 画图，45度线
   - 斜率 $\frac{d y_{t+1}}{d y_t}$
   - Dynamic：Page 23
     - stable / unstable: 斜率绝对值小于 1/大于 1.
     - converges/diverges:
     - oscillates/monotone：   斜率小于0/大于0   
  - IVP: initial value problem   

### 18.1 2017 midterm1 q2

### 18.2 2016 final 1

注意读懂题目。

### 18.3 2016 midterm1 q1


------------------------------------------

## 19. [Slides Chapter 19 - First Order Autonomous Nonlinear Difference Equations ](http://web.uvic.ca/~rondeau/econ351/protected/ch19slides.pdf)

- repeat and find the pattern
- converges or diverges,
- stabilize in the long run



### 19.1 2016 final q7
Nonlinear de usually do not have analytical solution
- steady state $\bar y$
- phase diagram
  - 45度线
  - 和45度线相交的地方，steady state
    - 看斜率，$ \frac{d y_{t+1}}{d y_t} $ evaluated at SS
- Dynamic：Page 14
  - stable / unstable: 斜率绝对值小于 1/大于 1.
  - converges/diverges:
  - oscillates/monotone：   斜率小于0/大于0   

### 19.2 2015 final q4
 - 注意总结规律，可以验算

### 19.3 2017 midterm1 5

### 19.4 2016 midterm1 q5


--------------------------------------

## 20. [Second Order Autonomous Linear Difference Equations - Slides Chapter 20](http://web.uvic.ca/~rondeau/econ351/protected/ch20slides.pdf)
- 分类： 2nd order Difference equations:

### 20.1 2016 mt2. Q4
* 基本情况
     - [RSolve[x[n] - x[1 + n] + x[2 + n] == 0, {x[n]}, n]](http://www.wolframalpha.com/input/?i=x%5Bn%2B2%5D+-+x%5Bn%2B1%5D+%2B+x%5Bn%5D+%3D0)


* 还是linear的，所以，区分homogeneous 和 particular solution两部分。
     - The most pragmatic approach is to proceed by dividing the equation into its particular and homogenous parts, proceed with an informed guess and see where that leaves us!
* 记住 characteristic equation，如果我们guess We guess: $y_t = Cr^t$。
  - 接着，就变成了解方程。
  - It is a quadratic with two roots, known as the characteristic roots or eigenvalues of the difference equation. The roots are given by the quadratic formula: P8
  - 三种情况，最后一种，最难，也最会考。但更可能考differential的。
    * $(h + vi) = R^t (cos \theta + i sin \theta )^t$
    * DeMoivre's Theorem
      $$ (cos \theta + i sin \theta )^t = (cos \theta t + i sin \theta t ) $$
  - 就是看看 p18

* Dynamic properties: Steady state  
  -  Case 1: Distinct Real Roots
     - Converges to its steady state only if $−1 < r1,r2 < 1$
     - Diverges if it does not have a steady state, or if either $|r1| > 1$ or$|r2| > 1$.
     - is monotone if both r1,r2 > 0
     - oscillates if either r1 < 0 or r2 < 0

  -  Case 3: with Complex Roots
     - This is a wave function/ oscillates
     - It converges to 0 if $0 < R < 1$
     - It diverges to ∞ if $R > 1$
     - It is constant and equal to 1 if a2 = 1 (我们这一题就是，看最后结果，也可以看出来)



   * 解完的情况，记得看看p31的完整的例子。
     - [RSolve[{x[2 + n] == -x[n] + x[1 + n], x[0] == 2, x[1] == 1}, x[n], {n}]](http://www.wolframalpha.com/input/?i=RSolve%5B%7Bx%5Bn%2B2%5D%3D%3Dx%5Bn%2B1%5D-x%5Bn%5D+,x%5B0%5D%3D%3D2,x%5B1%5D+%3D%3D1%7D,++++++++++++x%5Bn%5D,n%5D)


### 20.2 2017 midterm2 q4

------------------------------------------------



## 21. [First Order Linear Differential Equation - Chapter 21 ](http://web.uvic.ca/~rondeau/econ351/protected/ch21slides.pdf)


### 21.1 2016 midterm2 q4
 - separate
 - integrate
<commit_msg>update 350<commit_after>---
published: true
title: ECON351 FINAL 2017
category: econ351
tags:
  - econ351
  - Differential equations
---

重要的事情说三遍。

**Practice exam, Practice exam, Practice exam**

**刷题，刷题，刷题**

[What do top students do differently? | <NAME> |  
科学研究发现，
](https://www.youtube.com/watch?v=Na8m4GPqA30)

1. not IQ
2. not work hard
3. do Practice exam

少年，你离学霸的距离，就是少刷了一道题。

## 计划
1. 首先，把近些年 last final 做好。
2. 然后，复习近些年的midterm。
3. 最后，根据做错的题目，阅读slides。
4. 考试的时候，永远从最简单的题目，做起。不会的，难得，就留到后面。交卷前，一定检查。
-------------------------------------------------


## Homogenous and particular solution

练习例子
$$ \ddot y + a_1 \dot y + a_2 y = b$$



## 16.[Integration - Chapter 16 ](http://web.uvic.ca/~rondeau/econ351/protected/ch16slides.pdf)

### 16.1 2015 final q5

画图，帮助自己理解。


## 16.2 2016 MT2  Q1

Basic integration problem。 反求导。


[Integrate[10000 - 10 Sqrt[q] - q/10, {q, 0, 16}]](http://www.wolframalpha.com/input/?i=integrate(+10000+-+10*q%5E0.5+-+q%2F10,+0,+16))

[Plot[10000 - 10 Sqrt[q] - q/10, {q, 0, 16}]](https://sandbox.open.wolframcloud.com/app/objects/7972818e-79ce-4296-8b16-e17cd7295746#sidebar=compute)



----------------------------

## 18. [Slides Chapter 18 - First Order Autonomous Linear Difference Equations ](http://web.uvic.ca/~rondeau/econ351/protected/ch18slides.pdf)

the difference equation
 - solve
    - 不要背公式
    - 一步一步写到5步
    - 总结规律，写出close form
    - 转换成 $a^t(y_0 - \bar y)  + \bar y$
 - steady state. $\bar y$
 - the stability properties of the variable and steady state
   - 画图，45度线
   - 斜率 $\frac{d y_{t+1}}{d y_t}$
   - Dynamic：Page 23
     - stable / unstable: 斜率绝对值小于 1/大于 1.
     - converges/diverges:
     - oscillates/monotone：   斜率小于0/大于0   
  - IVP: initial value problem   

### 18.1 2017 midterm1 q2

### 18.2 2016 final 1

注意读懂题目。

### 18.3 2016 midterm1 q1


------------------------------------------

## 19. [Slides Chapter 19 - First Order Autonomous Nonlinear Difference Equations ](http://web.uvic.ca/~rondeau/econ351/protected/ch19slides.pdf)

- repeat and find the pattern
- converges or diverges,
- stabilize in the long run



### 19.1 2016 final q7
Nonlinear de usually do not have analytical solution
- steady state $\bar y$
- phase diagram
  - 45度线
  - 和45度线相交的地方，steady state
    - 看斜率，$ \frac{d y_{t+1}}{d y_t} $ evaluated at SS
- Dynamic：Page 14
  - stable / unstable: 斜率绝对值小于 1/大于 1.
  - converges/diverges:
  - oscillates/monotone：   斜率小于0/大于0   

### 19.2 2015 final q4
 - 注意总结规律，可以验算

### 19.3 2017 midterm1 5

### 19.4 2016 midterm1 q5


--------------------------------------

## 20. [Second Order Autonomous Linear Difference Equations - Slides Chapter 20](http://web.uvic.ca/~rondeau/econ351/protected/ch20slides.pdf)
- 分类： 2nd order Difference equations:

### 20.1 2016 mt2. Q4
* 基本情况
     - [RSolve[x[n] - x[1 + n] + x[2 + n] == 0, {x[n]}, n]](http://www.wolframalpha.com/input/?i=x%5Bn%2B2%5D+-+x%5Bn%2B1%5D+%2B+x%5Bn%5D+%3D0)


* 还是linear的，所以，区分homogeneous 和 particular solution两部分。
     - The most pragmatic approach is to proceed by dividing the equation into its particular and homogenous parts, proceed with an informed guess and see where that leaves us!
* 记住 characteristic equation，如果我们guess We guess: $y_t = Cr^t$。
  - 接着，就变成了解方程。
  - It is a quadratic with two roots, known as the characteristic roots or eigenvalues of the difference equation. The roots are given by the quadratic formula: P8
  - 三种情况，最后一种，最难，也最会考。但更可能考differential的。
    * $(h + vi) = R^t (cos \theta + i sin \theta )^t$
    * DeMoivre's Theorem
      $$ (cos \theta + i sin \theta )^t = (cos \theta t + i sin \theta t ) $$
  - 就是看看 p18

* Dynamic properties: Steady state  
  -  Case 1: Distinct Real Roots
     - Converges to its steady state only if $−1 < r1,r2 < 1$
     - Diverges if it does not have a steady state, or if either 绝对值$r1 > 1$ or 绝对值$r2 > 1$.
     - is monotone if both r1,r2 > 0
     - oscillates if either r1 < 0 or r2 < 0

  -  Case 3: with Complex Roots
     - This is a wave function/ oscillates
     - It converges to 0 if $0 < R < 1$
     - It diverges to ∞ if $R > 1$
     - It is constant and equal to 1 if a2 = 1 (我们这一题就是，看最后结果，也可以看出来)



   * 解完的情况，记得看看p31的完整的例子。
     - [RSolve[{x[2 + n] == -x[n] + x[1 + n], x[0] == 2, x[1] == 1}, x[n], {n}]](http://www.wolframalpha.com/input/?i=RSolve%5B%7Bx%5Bn%2B2%5D%3D%3Dx%5Bn%2B1%5D-x%5Bn%5D+,x%5B0%5D%3D%3D2,x%5B1%5D+%3D%3D1%7D,++++++++++++x%5Bn%5D,n%5D)


### 20.2 2017 midterm2 q4

------------------------------------------------



## 21. [First Order Linear Differential Equation - Chapter 21 ](http://web.uvic.ca/~rondeau/econ351/protected/ch21slides.pdf)


### 21.1 2016 midterm2 q4
 - separate
 - integrate

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