ECON351 MT2 2017

Posted by ECON爱好者 on March 16, 2017   econ351

2016 MT2

基本上，16，20，21，22, 23章都考了。就剩下最后第一个system equations，没有考。

老师特别提到要考， 23章，这是以前期中第二次，没有的情况。

看来，肯定会考到。

    1. Q1

Basic integration problem。 反求导。

Integration - Chapter 16

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

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

    1. Q2

Separable equation:

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

  • 2.1 ODE classification:
  • 2.2 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
  • 2.3 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

  • 2.4 看看走势图和 phase diagram。

  • 2.5 IVP (Initial value problem)
    • 看看maple 的 de plot。 direction field。 满满都是箭头。
    • 但是，一旦选定了起点，那么，就一条道走到黑。走到SS（steady state）

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。

    1. Q3
    • 3.1 ODE classification:
    • 3.2 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。
    • 3.3 Steady state 看看maple， 画的图。direction field。
    • Qualitatively analyze the diagram and the steady states

    • The Steady States The Steady States are the values ¯y such that y˙ = 0 - Identifying the general motion of the variable: 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.
    • 3.4 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

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

Slides Chapter 20 - Second Order Autonomous Linear Difference Equations 2nd order Difference equations:

  • 基本情况
  • 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，也就会考。
    • 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 - 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)

  • 解完的情况，记得看看p31的完整的例子。

https://reference.wolfram.com/language/tutorial/MakingDefinitionsForFunctions.html

Related Posts