//////////////////////////////////////////////////////// 

:05 2013-11-5 Tuesday 

ODE == Ordinary Differential Equations 

 

9:06 2013-11-5 

1st order ODE 

 

y' = f(x, y) 

 

9:34 2013-11-5 

geometric view of ODE's 

 

9:36 2013-11-5 

analytic view 

 

9:39 2013-11-5 

y' = f(x, y) <----> direction fields 

 

y(x) sol <----> integral curve 

 

9:43 2013-11-5 

integral curve 

 

10:50 2013-11-5 

isocline 

 

11:06 2013-11-5 

isocline, direction fields, integral curve 

 

11:06 2013-11-5 

solve ODE by separating variables 

 

11:27 2013-11-5 

lobster trap 

 

11:31 2013-11-5 

solution can not escape 

 

11:32 2013-11-5 

principle NO 1: 

 

2 integrals curves can not cross by an angle 

 

11:40 2013-11-5 

2 integrals can not be tangent! 


 

because EXISTENCE & UNIQUENESS THEOREM 

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

15:13 2013-11-5 

start 18.06 lec 2, Euler's numerical solution 

 

15:13 2013-11-5 

geometric solution, analytic solution, 

 

numetic solutions 

 

15:14 2013-11-5 

initial condition 

 

15:14 2013-11-5 

IVP == Initial Value Problem 

 

15:21 2013-11-5 

step size 

 

15:29 2013-11-5 

be systematic 

 

15:41 2013-11-5 

convex, concave 

 

15:53 2013-11-5 

DE == Differential Equations 

 

15:56 2013-11-5 

you're automatically introducing a 

 

systematic error! 

 

15:58 2013-11-5 

take small steps and it will follow better 

 

15:58 2013-11-5 

use small step size 

 

16:02 2013-11-5 

error e depends on the step size 

 

16:04 2013-11-5 


e ~ c1 * h // Euler is a first order method 

 

16:06 2013-11-5 

I halve the stepsize, I halve the error! 

 

16:10 2013-11-5 

however there are more efficient method 

 

that get the results more quickly! 

 

16:12 2013-11-5 

find a better slope! 

 

find a better value for An! 

 

16:22 2013-11-5 

improved Euler's method, 

 

Modified Euler's method, 

 

RK2 

 

16:23 2013-11-5 

RK2 is a 2nd order method 

 

e2 ~ c2 * h * h 

 

16:25 2013-11-5 

you have to evaluate the slope 4 times! 

 

in RK4 

 

16:27 2013-11-5 

RK4 is the standard method, it's not 

 

very efficient, but it's very accurate! 

 

16:28 2013-11-5 

What is the RK4 method? 

 

it's the Runge-Kutta method 4th order 

 

16:29 2013-11-5 

RK4 requires that you calculate 4 slopes: 


 

(An + 2*Bn + 2*Cn + Dn) / 6 // super slope 

 

16:34 2013-11-5 

singularity 

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

19:28 2013-11-5 

1st order linear equations 

 

19:34 2013-11-5 

constant coefficient 

 

19:34 2013-11-5 

homogeneous 

 

19:40 2013-11-5 

standard linear form 

 

19:45 2013-11-5 

temperature concentration model 

 

19:57 2013-11-5 

Newton's law of cooling 

 

19:59 2013-11-5 

Why "conduction model"? 

 

trasmit heat by conduction 

 

20:02 2013-11-5 

It's proportional to the temperature difference 

 

20:04 2013-11-5 

Newton's law of cooling 

 

dT/dt = k * (Te - T) 

 

20:07 2013-11-5 

external temperature, internal temperature 

 

20:09 2013-11-5 

the diffusion model 

 

20:10 2013-11-5 


salt concentration inside, 

 

salt concentration outside 

 

20:12 2013-11-5 

membrane wall 

 

20:13 2013-11-5 

equation is the same, except it got 

 

the name of diffusion equation 

 

20:15 2013-11-5 

internal concentration, external concentration 

 

20:16 2013-11-5 

this is the diffusion equation, and 

 

this is the conduction equation 

 

20:18 2013-11-5 

let's put it into "standard linear form" 

 

20:21 2013-11-5 

What is a standard linear form? 

 

y' + p(x)y = q(x) 

 

20:29 2013-11-5 

integrating factor 

 

20:31 2013-11-5 

What is an "integrating factor"? 

 

in order to solve: y' + p(x)y = q(x) 

 

using integrating factor: u(x) 

 

uy' + puy = qu, 

 

then uy' + puy == ( )' ??? probably 

 

20:50 2013-11-5 

separate variable 


 

20:50 2013-11-5 

indefinite integral 

 

20:51 2013-11-5 

we can find the integrating factor by 

 

using a formula! 

 

20:52 2013-11-5 

summarize it as a clearcut method 

 

21:00 2013-11-5 

I.F. == Integrating Factor 

 

21:20 2013-11-5 

Linear Differential Equation with Constant Coefficient 

 

21:20 2013-11-5 

LCCDE == Linear Constant-Coefficient Differential Equation 

 

21:28 2013-11-5 

engineer literature 

 

21:30 2013-11-5 

definite integral solutions 

 

indefinite integral solutions 

 

21:33 2013-11-5 

steady-state solution, transient solution 

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

21:54 2013-11-5 

start ODE lec 04, change variables, 

 

substitution in ODE 

 

21:56 2013-11-5 

change of variables(substitution) 

 

21:57 2013-11-5 

scaling 

 

22:02 2013-11-5 


internal temperature, external temperature 

 

22:04 2013-11-5 

dimensionless, 

 

 there is no unit attached to it 

 

22:10 2013-11-5 

they are lumped for you, and you just give 

 

the lump a new name 

 

22:11 2013-11-5 

we'll render unsolvable problems suddenly 

 

solvable 

 

22:11 2013-11-5 

there is two kind of substitution: 

 

1. direct ~ 

 

2. inverse ~ 

 

22:13 2013-11-5 

direct substitution, inverse substitution 

 

22:17 2013-11-5 

What is a direct substitution? 

 

a new varible is the combination of the old varialble 

 

22:18 2013-11-5 

What is a inverse substitution? 

 

old varialbe = f(new, old); 

 

22:29 2013-11-5 

Bernoulli Equations 

 

22:31 2013-11-5 

What is a Bernoulli equation? 

 

y' = p(x)y + q(x)exp(y, n) 


 

22:39 2013-11-5 

linear equation 

 

22:39 2013-11-5 

standard linear form 

 

22:45 2013-11-5 

Let's put into standard form! 

 

22:45 2013-11-5 

that's a linear equation that is in 

 

standard linear form 

 

22:52 2013-11-5 

arbitrary constant 

 

22:55 2013-11-5 

homogeneous 1st order ODEs 

 

22:56 2013-11-5 

homogeneous ODE 

 

22:58 2013-11-5 

What is a homogeneous ODE? 

 

y' = f(y/x) 

 

22:59 2013-11-5 

that's a homogenous equation, because I 

 

can see it can be written that way 

 

23:00 2013-11-5 

you can tuck inside the square root 

 

23:02 2013-11-5 

invariant under zoom 

 

23:03 2013-11-5 

I'm expanding equally, that's what I called 

 

zoom.. 


 

23:06 2013-11-5 

since I scale them equally 

 

23:07 2013-11-5 

identical equation 

 

23:10 2013-11-5 

direct substitution, inverse substitution 

 

23:14 2013-11-5 

beam of light, drug boat 

 

23:17 2013-11-5 

the angle between the beam & the boat 

 

23:19 2013-11-5 

What's the boat's path? 

 

23:20 2013-11-5 

the slope of the curve makes a constant angle 

 

23:20 2013-11-5 

differential equations 

 

23:32 2013-11-5 

homogeneous equation 

 

23:44 2013-11-5 

exponential spiral 

////////////////////////////////////////////////// 

10:07 2013-11-6 

1st order autonomous differential equation 

 

10:09 2013-11-6 

What is an autonomous ODE? 

 

dy/dt = f(y) // no t on RHS 

 

10:09 2013-11-6 

RHS == RightHandSide 

 

10:15 2013-11-6 

What does it mean by "autonomous"? 


 

autonomous == no independent variable on RHS 

 

10:19 2013-11-6 

to get qualitative solutions without 

 

having to solve the equation! 

 

10:20 2013-11-6 

How does the direction fields look? 

 

10:27 2013-11-6 

critical point 

 

10:39 2013-11-6 

integral curve 

 

10:49 2013-11-6 

continuous interest rate 

 

10:51 2013-11-6 

w is the rate of embezzlement 

 

10:53 2013-11-6 

I want to analyze this equation using 

 

the method of critical point! 

 

11:06 2013-11-6 

logistic equation(population equation) 

 

11:09 2013-11-6 

net birth rate(growth rate) 

 

11:10 2013-11-6 

if k is constant, then it's called a simple 

 

population growth 

 

11:16 2013-11-6 

partial fraction 

 

11:19 2013-11-6 

critical point method 


 

11:21 2013-11-6 

here is my dy/dt axis, and here is my y axis 

 

11:26 2013-11-6 

increasing aymptotically to the level curve a/b 

 

11:27 2013-11-6 

physically they do not mean anything, but 

 

mathematically they exist 

 

11:29 2013-11-6 

this is called a stable solution, because every 

 

tries closer & closer to it! 

 

11:29 2013-11-6 

unstable solution 

 

11:29 2013-11-6 

c.p. == critical point 

 

11:30 2013-11-6 

stable critical point 

 

unstable critical point 

 

11:32 2013-11-6 

there is a long-term solution 

 

11:35 2013-11-6 

How would the corresponding curve look? 

 

11:35 2013-11-6 

If I start below it, I rise to it 

 

If I start above it, I leave it! 

 

11:36 2013-11-6 

stable on one side, and unstable on the 

 

other side, it's called semi-stable 

 


11:40 2013-11-6 

logistic equation with harvesting 

 

11:42 2013-11-6 

harvest is at a constant-time rate 

 

11:59 2013-11-6 

so this curve has no critical point attached 

 

to it! 

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

12:14 2013-11-6 

review ODE lec 5, autonomous equations 

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

15:14 2013-11-6 

start ODE lec 6, Complex Numbers 

 

15:15 2013-11-6 

complex conjugate 

 

15:16 2013-11-6 

What is a complex conjugate? 

 

if z = a + bi, then its complex conjugate 

 

is z' = a - bi 

 

15:19 2013-11-6 

polar representation 

 

15:20 2013-11-6 

polar coordinate 

 

15:23 2013-11-6 

Euler's formula 

 

15:27 2013-11-6 

exponential function 

 

15:28 2013-11-6 

exponential law 

 

15:28 2013-11-6 

Why exponential function exp(e, x) is so popular? 


 

because the derivative of exp(e, x) is exp(e, x) 

 

itself 

 

15:35 2013-11-6 

infinite series 

 

15:39 2013-11-6 

using the method of grouping 

 

15:42 2013-11-6 

trignometric identity 

 

15:46 2013-11-6 

complex-valued funtion of a real-valued variable 

 

15:47 2013-11-6 

see exp(i*theta) as a blackbox system, then 

 

input == real, but output == complex! 

 

15:50 2013-11-6 

real part + imaginary part 

 

16:02 2013-11-6 

polar form of a complex number 

 

exp(a + ib) == exp(a) * exp(ib) 

 

16:05 2013-11-6 

r == modulus 

 

16:10 2013-11-6 

Why polar coordinate? 

 

because polar form is good for multiplication 

 

16:11 2013-11-6 

cartesian form, polar form 

 

16:14 2013-11-6 

How to multiply two complex numbers in polar form? 

 


1. multiply modulus 

 

2. add arguments 

 

16:18 2013-11-6 

integration by part 

 

16:40 2013-11-6 

cos(x) is the real part of e to the ix 

 

16:44 2013-11-6 

complex exponential 

 

16:53 2013-11-6 

find the nth root of 1 in the complex domain? 

 

16:55 2013-11-6 

here is the unit circle 

 

16:56 2013-11-6 

geometrically it's clear that these 

 

are the 5th roots of 1 

 

17:00 2013-11-6 

Is this the fifth root of 1? 

 

17:02 2013-11-6 

since 2pi and zero are the same angle 

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

17:21 2013-11-6 

start ODE lec 7, first order differential equations 

 

17:22 2013-11-6 

temperature concentration model 

 

17:25 2013-11-6 

integrating factor 

 

17:29 2013-11-6 

steady-state solution(long-term solution), 

 

transient solution 

 


17:32 2013-11-6 

SSS == Steady-State Solution 

 

17:47 2013-11-6 

the input is the RHS of the differential 

 

equation! 

 

17:48 2013-11-6 

the external water bath temperature 

 

17:50 2013-11-6 

driving external temperature 

 

17:51 2013-11-6 

response is the solution to the differential 

 

equation! 

 

17:57 2013-11-6 

physical input: Qe(t) 

 

17:57 2013-11-6 

superposition principle 

 

18:00 2013-11-6 

higher-order equations 

 

18:00 2013-11-6 

lineararity of the ODE 

 

18:01 2013-11-6 

What happening if input is trignometric? 

 

18:08 2013-11-6 

Ω is the angular frequency 

 

18:10 2013-11-6 

What is an angular frequency? 

 

number of complete oscillatins in 

 

2 pi distance 

 


18:19 2013-11-6 

complex exponential is the eigenfunction 

 

of LTI system 

 

18:25 2013-11-6 

the real part will solve the original ODE 

 

18:30 2013-11-6 

integrating factor 

 

18:37 2013-11-6 

polar coordinate, cartesian coordinate 

 

18:47 2013-11-6 

right triangle 

 

18:50 2013-11-6 

complex solution 

 

18:57 2013-11-6 

phase delay(phase lag) 

 

19:06 2013-11-6 

amplitude, phase lag 

 

///////////////////////////////////////////// 

9:11 2013-11-7 Thursday 

start ODE lec 08 

 

9:17 2013-11-7 

phase lag 

 

9:21 2013-11-7 

complex conjugate 

 

9:21 2013-11-7 

What is a complex conjugate? 

 

the complex conjugate of a + ib is 

 

a - ib, complex conjugates are symmetric 

 

about the x axis! 


 

9:26 2013-11-7 

trigonometric identity 

 

9:55 2013-11-7 

unit vector 

 

9:56 2013-11-7 

scalar product(dot product) 

 

10:16 2013-11-7 

linear equation 

 

10:17 2013-11-7 

basic linear ODE 

 

10:20 2013-11-7 

definite integral, indefinite integral 

 

10:21 2013-11-7 

first thing first! 

 

10:23 2013-11-7 

mixing example 

 

10:23 2013-11-7 

flow rate 

 

10:23 2013-11-7 

x(t): amount of salt in tank at time t 

 

10:26 2013-11-7 

Ce: concentration of the incoming salt 

 

10:29 2013-11-7 

rate of change of salt in the tank 

 

10:31 2013-11-7 

rate of change of salt inflow, 

 

rate of change of salt outflow 

 

10:35 2013-11-7 

concentration diffusion equation 


 

10:36 2013-11-7 

let's put it in standard form 

 

10:42 2013-11-7 

Newton's law of diffusion, 

 

Newton's law of cooling 

 

10:43 2013-11-7 

concentration model 

 

10:58 2013-11-7 

since the current throught the capacitor 

 

does not make sense, you have to talk charge 

 

on the capacitor! 

 

11:05 2013-11-7 

radioactive substance decay 

 

11:08 2013-11-7 

radioactive decay law 

 

11:14 2013-11-7 

initial condition 

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

11:50 2013-11-7 

start ODE, lec 09, linear 2nd order ODE 

 

11:56 2013-11-7 

constant coefficient 

 

11:57 2013-11-7 

homogeneous 

 

11:58 2013-11-7 

inhomogeneous 

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

16:50 2013-11-7 

start ODE lec 09 

 

///////////////////////////////////////////////// 


16:22 2013-11-8 Friday 

start ODE lec 09, 2nd order LCCDE 

 

16:22 2013-11-8 

dashpot 

 

16:23 2013-11-8 

spring-mass-dashpot 

 

17:05 2013-11-8 

characteristic equation 

 

17:28 2013-11-8 

heavily damped 

 

17:32 2013-11-8 

damping constant 

 

17:40 2013-11-8 

overdamped case 

 

17:43 2013-11-8 

complex roots 

 

17:52 2013-11-8 

real constant 

 

17:58 2013-11-8 

trigonometric identity 

 

18:19 2013-11-8 

underdamped case 

 

18:20 2013-11-8 

critically damped 

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

18:36 2013-11-8 

start lec 10, oscillations 



17:43 2013-11-8 

complex roots 

 

17:52 2013-11-8 

real constant 

 

17:58 2013-11-8 

trigonometric identity 

 

18:19 2013-11-8 

underdamped case 

 

18:20 2013-11-8 

critically damped 

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

18:36 2013-11-8 

start lec 10, oscillations 
 

18:36 2013-11-8 

oscillations corresponds to complex roots 

 

18:50 2013-11-8 

general solution 


 

19:08 2013-11-8 

complex conjugate 

 

20:27 2013-11-8 

damped case 

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

///////////////////////////////////////////////
21:25 2014-2-14
start ODE, lec 1, geometric view of ...

21:25 2014-2-14
1st order ODE

21:30 2014-2-14
y' = f(x, y)

21:30 2014-2-14
solve by separate variables

21:30 2014-2-14
geometric view of looking at differential equations

21:32 2014-2-14
analytic method

numeric method

geometric method

21:33 2014-2-14
y' = f(x, y)  <=> direction field

y(x) solution <=> integral curve

21:34 2014-2-14
integral curve of direction fields

21:44 2014-2-14
how to draw direction field?

21:50 2014-2-14
level curve    //isocline

21:54 2014-2-14
What is an isocline?

f(x, y) = c // slope at all points on the isocline are equal!



21:57 2014-2-14
isocline => direction fields => integral curve

22:04 2014-2-14
it's like a lobster trap

22:16 2014-2-14
two integral curve cannot cross!

// because you cannot have two different slope at a single point!

22:19 2014-2-14
two integral curves cannot be tangent(touch)!

22:22 2014-2-14
existence & uniqueness theorem

/////////////////////////////////////////////////////////////
16:35 2014-2-15 Saturday

ODE, lec 2, Euler's numeric method

16:35 2014-2-15
numeric solution

16:36 2014-2-15
IVP == Initial Value Problem

16:36 2014-2-15
IVP == Differential Equation + initial value

16:37 2014-2-15
Euler's method

16:37 2014-2-15
uniform stepsize which usually called h

16:41 2014-2-15
h // step size

16:44 2014-2-15
convex, concave

17:50 2014-2-15
the bigger the stepsize, the bigger the error!

17:55 2014-2-15
e ~ c1 * h // Euler is a 1st order method

17:58 2014-2-15
have the stepsize, have the error!

18:00 2014-2-15
better methods: 

find a better value for An (the slope)!

18:03 2014-2-15
improved Euler's method,

modified Euler's method,

RK2

18:13 2014-2-15
RK2 is a 2nd order method

18:13 2014-2-15
RK4 is the standard method, 

accurate but inefficient!

18:19 2014-2-15
Runge-Kutta 4th order method

18:24 2014-2-15
singularity  // singular point

18:24 2014-2-15
start ODE, lec 3, solving 1st order linear equation

18:27 2014-2-15
standard linear form for 1st order linear equation:

y' + p(x)y = q(x)

19:27 2014-2-15
models: 

temperature-concentration model

mixing model

19:28 2014-2-15
water bath

19:32 2014-2-15
Newton's law of cooling

19:33 2014-2-15
diffusion model

19:38 2014-2-15
external water bath

19:38 2014-2-15
salt concentration inside,

salt concentration outside

19:41 2014-2-15
dT/dt = k(Te - T)

19:44 2014-2-15
let's put it into standard linear form!

19:44 2014-2-15
integrating factor

19:49 2014-2-15
calculating integrating factor

21:38 2014-2-15
* linear with k constant

22:09 2014-2-15
steady-state solution,

transient solution

22:18 2014-2-15
start ODE, lec 4

22:19 2014-2-15
substitution // change of variables

22:19 2014-2-15
*linear equations

*equations with variables that are separable

22:21 2014-2-15
change of variables:

* scaling

22:25 2014-2-15
for big temperature difference, Newton's cooling law

breaks down, ...

22:33 2014-2-15
go to bed...

//////////////////////////////////////////////////
16:37 2014-2-17 Monday
ODE, lec 4, substitution in DE,

change of variables

16:37 2014-2-17
* scaling

16:41 2014-2-17
external temperature, internal temperature

16:48 2014-2-17
lumping constant

16:55 2014-2-17
direct substitution

inverse substitution

16:57 2014-2-17
Bernoulli equation => linear equation  // direct substitution

17:07 2014-2-17
integrating factor

17:16 2014-2-17
homogeneous 1st order ODEs

17:23 2014-2-17
"invariant under zoom"

17:29 2014-2-17
direct subsitution <=> inverse substitution

17:38 2014-2-17
light house, beam, drug boat

17:43 2014-2-17
start ODE, lec 05

1st-order autonomous ...

18:08 2014-2-17
autonomous: no independent varialbe on the RHS 

18:08 2014-2-17
qualitative information

20:03 2014-2-17
critical point

20:07 2014-2-17
analyze the equation using the method of critical point

20:20 2014-2-17
logistic equation  // population equation

population behavior

20:32 2014-2-17
dy/dt = k * y  // k is the growth rate

20:33 2014-2-17
k constant: simple growth

k is not constant: logistic growth

20:33 2014-2-17
c.p. == critical point

20:42 2014-2-17
stable critical point,

unstable critical point

20:48 2014-2-17
logistic equation with harvesting

20:55 2014-2-17
maximum rate of harvesting

21:07 2014-2-17
start ODE, lec 06, complex numbers

21:08 2014-2-17
complex conjugate

21:17 2014-2-17
polar representation of complex numbers

21:21 2014-2-17
exponential law // law of exponents

21:25 2014-2-17
exponential function

21:26 2014-2-17
infinite series

21:30 2014-2-17
the method of grouping

21:33 2014-2-17
trigonometric identity

21:36 2014-2-17
complex-valued function of a real variable

21:44 2014-2-17
polar form of a complex number

rectangular form of a ~

21:57 2014-2-17
r == modulus

theta == argument

22:00 2014-2-17
advantage of polar form is that it's good

for multiplication

22:01 2014-2-17
real part, imaginary part

22:14 2014-2-17
calculating the nth roots of 1

22:21 2014-2-17
ODE lec 07, 1st-order linear

/////////////////////////////////////////////////
17:57 2014-2-18
steady-state solution // long term solution

17:57 2014-2-18
input => response

18:03 2014-2-18
steady-state solution  // newton's law of cooling

18:04 2014-2-18
physical input == trigonometric functions

18:08 2014-2-18
input, response

19:41 2014-2-18
complexify the problem

19:42 2014-2-18
driving frequency, driving angular frequency

20:05 2014-2-18
phase lag // phase delay

20:05 2014-2-18
response function

20:11 2014-2-18
signals & systems view of ODE

20:11 2014-2-18
SSS == Steady-State Solution

22:01 2014-2-18
input(drive): temperature of the water bath

output(response): internal temperature

22:06 2014-2-18
the principle of superposition

22:11 2014-2-18
linearity of the ODE

22:12 2014-2-18
what if the input == trignometric function?

22:14 2014-2-18
angular frequency(circular frequency)

22:15 2014-2-18
start ODE, lec 08, continuation...

22:17 2014-2-18
transport the problem into the complex domain

22:17 2014-2-18
polar method <-> Cartesian method

22:20 2014-2-18
same frequency, with changed amplitude & phase shift

22:39 2014-2-18
basic linear ODE

22:57 2014-2-18
Temperature-Concentration Model

Conduction-Diffusion Model

22:57 2014-2-18
Mixing problem

23:02 2014-2-18
flow rate in,

flow rate out

23:03 2014-2-18
x(t): amout of salt in the tank at the time t

23:03 2014-2-18
amount of salt

23:06 2014-2-18
incoming concentration

23:10 2014-2-18
diffusion model <=> mixing model

23:23 2014-2-18
radioactive chain decay

23:32 2014-2-18
go to bed

/////////////////////////////////////////////
14:39 2014-2-19
start ODE, lec 09,

LCCDE 2nd-order

14:40 2014-2-19
inhomogeneous

14:42 2014-2-19
spring-mass-dashpot system

14:47 2014-2-19
characteristic equation of the ODE

15:01 2014-2-19
damping constant, spring constant

15:05 2014-2-19
initial condition, 

IVP == Initial Value Problem

15:09 2014-2-19
this is heavily damped

15:09 2014-2-19
equilibrium position

15:33 2014-2-19
overdamped

15:35 2014-2-19
complex roots

15:35 2014-2-19
stiff spring

15:47 2014-2-19
underdamped case

15:56 2014-2-19
critical damped

16:07 2014-2-19
ODE, lec 10 continuation of ....

16:18 2014-2-19
oscillations

16:18 2014-2-19
oscillations associates with complex roots

16:19 2014-2-19
oscillations <=> complex roots

16:36 2014-2-19
complex conjugate

16:48 2014-2-19
complex => oscillations

17:08 2014-2-19
harmonic motion  // undamped case

17:11 2014-2-19
it's crossing the axis periodically

17:25 2014-2-19
start ODE, lec 11, theory of general 2nd-order

LCCDE

17:38 2014-2-19
superposition principle

17:53 2014-2-19
linear homogeneous ODE

17:53 2014-2-19
differentiation operator

17:54 2014-2-19
I'm applying D to y! not multiply D with y!

17:55 2014-2-19
linear operator

17:56 2014-2-19
how to solve homogeneous differential equations?

take a system view, if the output == 0, what should

I put in as an input?

17:58 2014-2-19
linear combination

18:04 2014-2-19
fit initial conditions

18:11 2014-2-19
a pair of simultaneous linear equations

18:20 2014-2-19
Wronskian

18:35 2014-2-19
finding normalized solution

18:35 2014-2-19
sinh(x) // hyperbolic sine of x

cosh(x) // hyperbolic cosine of x

18:49 2014-2-19
why did engineers like normalized solutions?

18:50 2014-2-19
solving simultaneous linear equations

18:56 2014-2-19
existence & uniqueness theorem

21:09 2014-2-19
start ODE lec 12, continuation...

21:10 2014-2-19
inhomogeneous LCCDE

21:10 2014-2-19
input signal (driving term)(forcing term)

21:11 2014-2-19
response // the solution

21:11 2014-2-19
associated homogeneous equation

21:13 2014-2-19
reduced equation

21:13 2014-2-19
external force

21:21 2014-2-19
forced system <-> passive system

passive system // homogeneous system

21:26 2014-2-19
capacitance(C), inductance(L)

21:28 2014-2-19
the passive circuits without put extra electromagnetic force

21:35 2014-2-19
Ly = f(x) // L is the linear operator

 // inhomogeneous equation

21:38 2014-2-19
y = yp + yc, 

yp is the particular solution

yc is the complementary solution,// == yh, homogeneous solution

21:41 2014-2-19
linear operator

21:47 2014-2-19
inhomogeneous differential equations

21:49 2014-2-19
linear 1st-order equations

21:56 2014-2-19
associated homogeneous solution

21:59 2014-2-19
transient solution + steady-state solution

22:02 2014-2-19
ODE is stable, if the complementary goes to zero

22:11 2014-2-19
SSS == Steady-State Solution

22:11 2014-2-19
characteristic roots: roots of the characteristic equation

22:13 2014-2-19
stability condition

22:14 2014-2-19
stability condition:

the ODE is stable if the characteristic roots 

have nagative real part!

22:21 2014-2-19
start ODE, lec 13, find particular solutions(Xp)

22:28 2014-2-19
general solution = particular solution + complementary solution

22:51 2014-2-19
the rule of substitution

22:58 2014-2-19
polynomial operator

22:58 2014-2-19
p(D) // simple quadratic polynomial

22:59 2014-2-19
input signal == complex exponential

23:00 2014-2-19
D // the differentiation operator

23:01 2014-2-19
exponential-input theorem

23:05 2014-2-19
decaying oscillation

23:13 2014-2-19
general solution = complementary solution + particular solution

23:14 2014-2-19
let's complexify it

23:15 2014-2-19
imaginary part of the complex exponentials

23:15 2014-2-19
complex solution to this complexified equation

23:28 2014-2-19
exponential-shift rule

23:28 2014-2-19
product rule

23:34 2014-2-19
D == the differentiation operator

23:34 2014-2-19
what if a is a double-root?

0:13 2014-2-20
go to bed

////////////////////////////////////////////////////
13:38 2014-2-20
start ODE, lec 14

13:38 2014-2-20
resonance

13:38 2014-2-20
driving term(forcing term)

13:42 2014-2-20
the driving frequency is different from the natrual frequency

13:43 2014-2-20
driving frequency(input frequency)

natrual frequency

13:43 2014-2-20
complexify the equation

13:44 2014-2-20
input => response

13:49 2014-2-20
driving frequency, natrual frequency,

resonance!

13:50 2014-2-20
complementary solution

14:08 2014-2-20
Backward Euler

14:21 2014-2-20
beats

14:30 2014-2-20
damped resonance

14:51 2014-2-20
natrual undamped frequency

14:52 2014-2-20
natrual damped frequency(pseudo-frequency)

14:55 2014-2-20
damped spring

14:58 2014-2-20
start ODE, lec 15

introduction to Fourier series

15:04 2014-2-20
why Fourier series?

y'' + ay' + by = f(t)  // f(t) is the input

15:09 2014-2-20
f(t): the basic input is exponential, sine, cosine

15:11 2014-2-20
input:

response:

15:22 2014-2-20
why can I use a superposition principle?

because the ODE is linear

15:22 2014-2-20
orthogonal relations

15:50 2014-2-20
u(t) & v(t) are both periodic with 2 * pi, 

they are orthogonal if....

15:54 2014-2-20
integration by parts

16:15 2014-2-20
orthogonality relations

16:47 2014-2-20
start ODE, lec 16 continuation more Fourier series

17:04 2014-2-20
even function, odd function => Fourier coefficient

17:23 2014-2-20
the uniqueness of Fourier series

17:25 2014-2-20
if f(t) is even, simplify the calculation of the 

coefficient 

17:29 2014-2-20
start ODE, lec 17

finding particular ...

18:05 2014-2-20
the mathematic basis for hearing

18:05 2014-2-20
square wave

18:16 2014-2-20
Fourier expansion

18:22 2014-2-20
associated homogeneous solution(reduced solution)

18:31 2014-2-20
driving frequency

18:36 2014-2-20
Fourier analysis is done by using resonance

19:12 2014-2-20
start ODE lec 19, introduction to Laplace transform

23:12 2014-2-20
go to bed

////////////////////////////////////////////////////////////
12:11 2014-2-21 Friday
start ODE, lec 18, Lapalace transform

12:11 2014-2-21
where does Laplace transform come from?

12:12 2014-2-21
power series

12:12 2014-2-21
continuous analog

12:23 2014-2-21
what's the difference between transform & operator?

transform: f(t) -> F(s)

operator:  f(t) -> g(t)

12:36 2014-2-21
linear transform

12:36 2014-2-21
kernel

13:10 2014-2-21
exponential-shift formula

13:17 2014-2-21
backwards Euler formula

13:20 2014-2-21
start ODE, lec 20,

using Laplace transform to solve LCCDE

13:43 2014-2-21
exponential type

13:46 2014-2-21
rapidly growing exponential

13:46 2014-2-21
derivative formula for Laplace transform

16:01 2014-2-21
integrate by parts

16:08 2014-2-21
f(t) is of exponential type

16:11 2014-2-21
start ODE lec 21, 

convolution formula for Laplace transform

16:14 2014-2-21
convolution integral,

convolution sum

16:16 2014-2-21
where does Laplace transform come?

Laplace transform is a continuous analog of power series

16:29 2014-2-21
double integral

16:46 2014-2-21
radioactive dumping

17:08 2014-2-21
example of convolution in practice

17:08 2014-2-21
radioactive waste

17:08 2014-2-21
dump rate

17:09 2014-2-21
dump rate * decay rate

17:25 2014-2-21
start ODE, lec 22, using Laplace transform

18:20 2014-2-21
why people like Laplace transform?

because it handles jump discontinuity very nicely

18:20 2014-2-21
unit box

18:24 2014-2-21
unit step function

18:39 2014-2-21
start ODE, lec 23, use with impulse input

18:42 2014-2-21
unit impulse

18:42 2014-2-21
impulse of a function f(t) within the integral

over [a, b]

19:16 2014-2-21
the impulse is the area under this curve

19:17 2014-2-21
unit step is not a differentiable

21:19 2014-2-21
equilibrium point

22:03 2014-2-21
transfer function: W(s)

22:11 2014-2-21
the weight function of the system: w(t)

22:12 2014-2-21
what w(t) really ?

it's "unit impulse response"

//////////////////////////////////////////////////