ECON351 FINAL 2017 - ECON爱好者

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? | Douglas Barton |
科学研究发现，

not IQ
not work hard
do Practice exam

少年，你离学霸的距离，就是少刷了一道题。

计划

首先，把近些年 last final 做好。
然后，复习近些年的midterm。
最后，根据做错的题目，阅读slides。
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

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}]

Plot[10000 - 10 Sqrt[q] - q/10, {q, 0, 16}]

18. Slides Chapter 18 - First Order Autonomous Linear Difference Equations

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
- 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 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。题目难懂。万变不离其宗。读不懂，不要紧。老师还没有离（xie）奇（e）到，接着就过不下去了。

看看b问，就知道读不懂不影响b问以后的答题。

所以，遇到难题绕道走，就是了。完了，检查的时候，再回来。

不过，有时候。道理，我都懂，可还是过不好这一生/过不了这一题。

a问，其实是考203呢！350教科书上也有这个例子。就是为了说明difference equation在经济学中应用很普遍。

b问的答案，简直就是九阴真经啊。完全通用到所有的ODE问题。建议，抄下来，背诵。很经典的回答方式。

回答这一题，大概20min。

18.3 2016 final q5

18.3 2016 midterm1 q1

19. Slides Chapter 19 - First Order Autonomous Nonlinear Difference Equations

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

19.1 2015 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
  回答这一题，大概10min到20min

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

分类： 2nd order Difference equations:

20.0 2017 midterm2 q1a

What is the general form of the solution that must be guessed for a homogenous second order autonomous linear difference equation?

$y_t^h = C r^t$

20.1 2016 mt2. Q4

基本情况
- RSolve[x[n] - x[1 + n] + x[2 + n] == 0, {x[n]}, n]
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 $r2 < 0$
  - 这些根本就记不住。
  - 不要紧，如果已经解出了final complete solution。
  - 一看就知道了。看看例题2017 midterm2 q4
- 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
  - (2016 mt2. Q4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}]

这一题，大概要30min

20.2 2017 midterm2 q4

大概20min。清楚简单。 ————————————————

21. First Order Linear Differential Equation - Chapter 21

这一部分，是后面所有内容的基础。

21.1 2016 midterm2 q2

separate
integrate

Separable equation:

$(k'(t))/(60 - 0.1 k(t)) = 1$

看看基本情况 http://www.wolframalpha.com/input/?i=dk%2Fdt+%3D+-0.1*k%2B+60,+k(0)+%3D+200

21.1.1 ODE classification:
- first-order linear ordinary differential equation First Order Linear Differential Equation - Chapter 21
  思路就是separate and integration.
21.1.2 步骤，就是因为是linear ，所以先分为 homogeneous 和 particular solution两部分。原理，看论证。
- particular solution：steady states are given by a rate of change equal to zero:$ y˙ = 0$
- homogeneous： separate and integration
- Adding together the Particular and Homogenous Solutions
21.1.3 Dynamic Properties p18
- First Order Linear Autonomous Ordinary Differential Equations are ALWAYS MONOTONE
- If $a > 0$, $e^−at → 0$ as $t → ∞$. The variable converges to itssteady state and the steady state is stable.
- If $a < 0$, $e^−at → ∞$ as $t → ∞$. The variable diverges +∞ or −∞ depending on the sign of C
21.1.4 看看走势图和 phase diagram。
21.1.5 IVP (Initial value problem)
- 看看maple 的 de plot。 direction field。满满都是箭头。就像人生，永远有那么多选择。我都不知道该怎么选了。不要紧。反正终点都是一样。
- 但是，一旦选定了起点，那么，就一条道走到黑。走到SS（steady state）
- 所以，你的问题，就是选好起点。不然，有些人就得，走得漫长一些。

DSolve[{k’[t] == 60 - 0.1 k[t], k[0] == 200}, k[t], t]

这一章，是所有后面的基础。

这一题，大概20min

22.First Order Nonlinear Differential Equation - Chapter 22

Nonlinear 一般都没有解析解。就画图，看看dynamics。只有特殊的BernoulliDifferentialEquation，靠着Bernoulli的聪明才智，我们才能把nonlinear变成linear的。后面就和1st order一样解决了， separately integrate。

22.1 2017 midterm q5

吃果果的送分题。

22.2 2016 mt2. Q3

22.2.1 ODE classification:
- first-order nonlinear ordinary differential equation
- First Order Nonlinear Differential Equation - Chapter 22
22.2.2 solve nonlinear
- Our objective remains to characterize the evolution of $y(t)$ over time
- Most nonlinear differential equations DO NOT have a known algebraic solution $y(t) = G(t)$ (i.e. are not easily integrated) but even when they do, it is often useful to have a visual representation of their behavior
- We proceed with an analysis of the Phase Diagram of y。
- 看看maple，画的捕捉之前的，和之后的phase diagram
- 类似于，382， 381，打鱼，砍树，杀猪的故事。
22.2.3 Steady state 看看maple，画的图。direction/vector field。
- Qualitatively analyze the diagram and the steady states
  - The Steady States The Steady States are the values $\bar y$ such that $\dot y = 0$
  - Identifying the general motion of the variable: $\dot y > 0$ indicates that the variable has a positive rate of time change. It is increasing over time
- The Stability Properties of the Steady State
  - Formally, a linear approximation of $g(y)$, measured at the steady state
  - P13 If gradient $g(y) < 0$, the variable converges to the steady state. It is stable
  - If gradient $g(y) > 0$, the variable diverges away from the steady state. It is unstable
- nonlinear FIRST ORDER AUTONOMOUS differential equations cannot oscillate. These variables are always monotone.
22.2.4 solve nonlinear 的特殊方法

方法特殊，回到我们的亲爱的 Bernoulli

Bernoulli’s equation: p’(t) = 0.14 p(t) - 0.0005 p(t)^2

什么是 Bernoulli’s equation

主要思路，就是change of variables. 把nonlinear的问题，转换成linear的问题。

看p21，两边都除以 $p^n$, 就是把最 nonlinear的部分，除掉。

然后，设置

$x = p^{1-n}$

会发现，变成了linear的。那么问题就回到了第二题，first-order linear ordinary differential equation。解完x，再回到p就可以了。

题外话。如果，还是non autonomous。接着可以

什么是 [integration factor](http://mathworld.wolfram.com/IntegratingFactor.html)

这里，没有这么多问题。

Mathematica

这一题，大概30min。

这后面都是前面midterm没有考过的。老司机也要注意开车了。

23. Second Order linear Differential Equation - Chapter 23

也算必考内容之一了。

23.1 2017 midterm2 q1b,c

What is the general form of the solution that must be guessed for a homogenous second order autonomous linear Differential equation?

$y_t^h = C e^{rt}$

In looking for a particular solution to $\ddot y + a_1 \dot y + a_2y = b$ , you find that there is no steady state. What is your next guess?

如果$a_2 =0$ $y_t^p = kt$

后面还有如果$a_1 = a_2 =0$

$y_t^p = kt^2$

Euler’s equation $e^{vti} = cos(vt) + i sin(vt)$

23.2 2015 final q3

简单，明确。非常典型。

particular solution: steady state
- 如果没有就要猜 $y_t^p = kt$或者 $y_t^p = kt^2$
Homogenous solution
- Guess $y(t) = A e^{rt} $
  - 看看和difference equation的区别。
- characteristic function
  - 解方程
- roots
  - Distinct
  - same
  - Complex roots
    - $e^{vti} = cos(vt) + i sin(vt)$
    - 记得要合并到最终结果
    - 一看，就知道h起到了关键的指导作用。
Dynamic
- $h<0$, stable, converges
IVP initial value problem
- 带入数字，解方程组

23.3 2017 midterm2 q3

这一题是特例。记不住特例的话，就麻烦。

第一种方法，把nonlinear变成 linear

第二种方法，就是

Homogenous solution
- Guess $y(t) = A e^{rt} $
- characteristic function
- roots
  - Distinct
  - same
  - Complex roots
    - $e^{vti} = cos(vt) + i sin(vt)$
particular solution: steady state
- 如果没有就要猜 $y_t^p = kt$或者 $y_t^p = kt^2$
Dynamic
- $h<0$, stable, converges
IVP initial value problem
- 带入数字，接方程组

23.2 2016 final q6.b

答案写得很清楚。

这一题，大概30min。很多题，够大家刷了。

24. Systems of Differential Equations - Chapter 24

Visualization 可以看看我给ECON501同学，画的兔子和狐狸，system of differential equations. 一个是兔子的增长规律，一个是狐狸的增长规律，和最后的solution 兔子和狐狸

还有就是和老师slides类似的几个例子solution

24.1 Substitution method(必考)

24.1.2 2015 final q1 和q2

相对简单，先看看，熟悉一下。

$\frac{d \dot y}{ d t} = \ddot y$

先求导，再substitute。

complete closed form solution
- Homogenous
  - Guess the solution $k e^{rt}$。记得k这里是vector。
    - 这里可以看一下什么是eigenvalues和eigenvectors. 看看linear algebra
  - solve $A-Ir = 0$,
    - eigenvalues, eigenvectors
    - set $k_{11} = 1$
    - 把k vector求出来。
  - linear combination of two solutions 这一题，没有要求particular solution。

24.1.1 2016 final q6.1

Transform the system of two differential equations above into a single linear second order differential equation for $y$
$\frac{d \dot y}{ d t} = \ddot y$

24.2 Direct method: the system in matrix form(必考)

24.2.1 2016 final q2 q6

这一题非常典型。

答案也写得很完整，体现了老师的匠心，不知道是什么星座的。

complete closed form solution
- Homogenous
  - Guess the solution $k e^{rt}$。记得k这里是vector。
    - 这里可以看一下什么是eigenvalues和eigenvectors.
  - solve $A-Ir = 0$,
    - eigenvalues, eigenvectors
    - set $k_{11} = 1$
  - linear combination of two solutions
- particular
  - set $\dot x = 0$ and $\dot y = 0$
  - solve steady state. 解方程组
- complete solution
Analyze the dynamics of the model
- 画图，先从SS 和 isocline开始。
  - 然后是四组上下左右箭头。
  - 接着可以加上vector field。
  - 每一点上的斜率 $slope = \frac{\dot y}{\dot x}$
- properties: slides Chapter 24 page47:
  - source: $0 < r1 < r2$
  - sink: $r1 < r2 < 0$
  - saddle point: $r_1>0$ and $r_2 <0$
- steady state point
- Isocline: set $\dot x = 0$ and $\dot y = 0$
- vector field : directions in 4 quadrant
  - $ \frac{d \dot x}{y} >< 0 $ 往左？往右？
  - $\frac{d \dot y}{x} >< 0$ 往上？往下？
- converges
- oscillate

这一题估计要30+min。

24.2.2 2016 final q6

24.3 Non-Linear Systems of Differential Equations （有必要会）

24.3.1 2015 final q6

老师答案很完整，必须学习。

Linear Approximation at the Steady States
J is the Jacobian matrix of the linearized system
- 什么是Jacobian，
- 什么是eigenvalue 和 eigenvector of a matrix ，看看linear algebra
有了J以后，就成了一个linear system和前面一样做了。
- J 就成原来的A. 我们关注的是在SS附近的Dynamic
- 所以，要SS带入到J 里面，就成了我们的A.
- 接着就是解方程。
The stability properties of the steady state can be determined as before, by finding the roots of J and referring to Table 1
- table 1 在slide24 p47.
- 记不住，也不要紧。看最终结果，可以猜出来。
Dynamic和phase diagram就和linear system一样了。这一题是超难的。估计40min。

计划

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。