18.03: Dierential Equations, Spring, 2006 Driving through the dashpot
The Mathlet Amplitude and Phase: Second order considers a spring/mass/dashpot system driven through the spring. If y (t) denotes the displacement of the plunger at the top of the spring
18.03 Class 38, May 15 Nonlinear systems: Jacobian matrices [1] The Nonlinear Pendulum.
The bob of a pendulum is attached to a rod, so it can swing clear around the pivot. This system is determined by three parameters: L m g length of pendulum
mass of bo
18.03 Class 37, May 12 Introduction to general nonlinear autonomous systems. [1] Recall that an ODE is "autonomous" if and not on t: x' = g(x) x' depends only on x
For example, I know an island in the St Lawrence River in upstate New York where there are
18.03 Muddy Card responses, May 10, 2006
1. So what good are exponential matrices? It seems to me that they dont allow us to skip any steps: it looks like you still have to calculate eigenvalues and then eigenvectors, and then use those calculations to co
18.03 Class 36, May 10 Review of matrix exponential Inhomogeneous linear equations [1] Prelude on linear algebra: AB.
If A and B are matrices such that the number of columns in A is
the same as the number of rows in B , then we can form the "product mat
18.03 Class 35, May 8 The companion matrix and its phase portrait; The matrix exponential: initial value problems. [1] We spent a lot of time studying the second order equation x" + bx' + kx = 0
and if b and k are nonnegative we interpreted them as the
18.03 Class 34, May 5 Classification of Linear Phase Portraits The moral of today's lecture: Eigenvalues Rule (usually) A is
[1] Recall that the characteristic polynomial of a square matrix p_A(lambda) In the 2x2 case p_A(lambda) where = det(A  lambda
18.03 Class 33, May 3 Complex or repeated eigenvalues [1] The method for solving u' = Au that we devised on Monday is this:
(1) Write down the characteristic polynomial A) p_A(lambda) = det(A  lambda I) = lambda^2  (tr A)lambda +(det lambda_2 v such tha
18.03 Class 32, May 1 Eigenvalues and eigenvectors [1] Prologue on Linear Algebra. [a b ; c d] [x ; y] = x[a ; c] + y[b ; d] :
Recall
A matrix times a column vector is the linear combination of the columns of the matrix weighted by the entries in the colu
18.03 Class 31, April 28, 2006 First order systems: Introduction [1] There are two fields in which rabbits are breeding like rabbits. Field 1 contains x(t) rabbits, field 2 contains y(t) rabbits. In both fields the rabbits breed at a rate of 3 rabbits per
18.03 Class 29, Apr 24 Laplace Transform IV: The pole diagram
[1] I introduced the weight function = unit impulse response with the mantra that you know a system by how it responds, so if you let it respond to the simplest possible signal (with the simple
18.03 Muddy Card responses, April 21, 2006
1. LTI = Linear, Time Invariant. This is a property of an operator or a system. An operator (which is a rule L that converts one function of time to another one) is linear if L(f + g ) = L(f )+ L(g ) and L(cf ) =
18.03 Muddy Card responses, April 14, 2006
1. A number of people brought up the point made at the end of Lecture 25, on April 12: how do we know what initial conditions yield the unit step or impulse responses? This is a tricky point and I did not explain
18.03 Lecture 26, April 14 Laplace Transform: basic properties; functions of a complex variable;
poles
diagrams; sshift law.
[1] The Laplace transform connects two worlds:
 The t domain 
 
 t is real and positive

 
 functions f(t) are
18.03 Class 25, April 12, 2006 Convolution [1] We learn about a system by studying it responses to various input signals. I claim that the weight function w(t)  the solution to p(D)x = delta(t) with rest initial conditions  contains complete data about
18.03 Class 24, April 10, 2006 Unit impulse and step responses [1] In real life one often encounters a system with unknown system parameters. If it's a spring/mass/dashpot system you may not know the spring constant, or the mass, or the damping constant.
18.03 Class 23, April 7 Step and delta. Two additions to your mathematical modeling toolkit.  Step functions [Heaviside]  Delta functions [Dirac] [1] Model of on/off process: a light turns on; first it is dark, then it is light. The basic model is the H
18.03 Class 22, April 5 Fourier series and harmonic response [1] My muddy point from the last lecture: I claimed that the Fourier series for f(t) converges wherever $f$ is continuous. What does this really say? For example, (pi/4) sq(t) for any value of t
18.03 Muddy Card responses, April 3, 2006
1. A number of people were confused by my derivation of the Fourier coecients of the function f (t), even, periodic, period 2 , with f (t) = 4 for 0 < t < /2 and f (t) = 0 for /2 < t < . I think the process of exp
18.03 Class 21, April 3 Fun with Fourier series [1] If f(t) is any decent periodic of period 2pi, it has exactly one expression as (*)
f(t) = (a0/2) + a1 cos(t) + a2 cos(2t) + . + b1 sin(t) + b2 sin(2t) + .
To be precise, there is a single list of coeffic
18.03 Class 20, March 24, 2006 Periodic signals, Fourier series [1] Periodic functions: for example the heartbeat, or the sound of a violin, or innumerable electronic signals. I showed an example of violin and flute. A function f(t) is "periodic" if there
18.03 Class 18, March 20, 2006 Review of constant coefficient linear equations: Big example, superposition, and Frequency Response [1] Example. x" + 4x = 0 PLEASE KNOW the solution to the homogeneous harmonic oscillator x" + omega^2 x = 0 are sinusoids of
18.03 Class 17, March 17, 2006 Application of second order frequency response to AM radio reception with guest appearance by EECS Professor Jeff Lang. [1] The AM radio frequency spectrum is divided into narrow segments which individual stations are requir
18.03 Class 16, March 15, 2006 Frequency response
[1] Frequency response: without damping
x" + omega_n^2 x = 0 :
First recall the Harmonic Oscillator: The spring constant is k = omega_n^2 .
Solutions are arbitrary sinusoids with circular frequency the "
18.03 Class 15, March 13, 2004 Operators: Exponential shift law Undetermined coefficients [1] Operators. D e^cfw_rt so and or D^n e^cfw_rt = = The ERF is based on the following calculation: r e^cfw_rt = rI e^cfw_rt
r^n I e^cfw_rt
(a_n D^n + . + a_0 I) e^c
18.03 Muddy Card responses, March 10, 2006
1. The commonest question concerned the idea and utility of operators. I'll say something now. You can look ahead at the "exponential shift law" if you want, to see one use later. An operator modifies a function
18.03 Class 14, March 10, 2006 Exponential signals, higher order equations, operators [1] Exponential signals x" + bx' + kx = A e^cfw_rt (*)
We want to find some solution. Try for a solution of the form k] b] xp xp' = = B e^cfw_rt xp = B e^cfw_rt :
B r e^
18.03 Class 13, March 8, 2006 Summary of solutions to homogeneous second order LTI equations; Introduction to inhomogneneous equations. [1] We saw on Monday how to solve x" + bx' + kx = 0.
Here is a summary table of unforced system responses. One of three