Linear phase portraits a 2 The matrices I want you to study all have the form A. 2 1 Compute the trace, determinant, c ha racteristic polynomial, and eigen v alues, in terms.

 

Euler s method; Linear models 1. Use Euler s method to estimate the value at x 1:5 of the solution of y0 y2 x2 F x; y at y 0 1. Use h 0:5. Recall the notation x0 0, y0 1, xn 1 xn h, yn 1 yn mnh, mn F xn;yn. Make a table with.

 

In tro duction, natural growth and decay, review of logarithm Tasks: 2. Suppose a 0. What is the doubling time 4. Check that the proposed solution satis.

 

Laplace transform II Rules for the Laplace transform 1 De nition: L f t F s f t e dt for Re s 0 0 Linearity: L af t bg t aF s bG s L 1: F s essen tially determines f t s -shift rule: L ertf t F s L tf t F 0 s t-derivative rule: L f0 t sF s ; L f n t snF s

 

6 F ourier Series: Intro duction 1. What is the general solution to x !2 x 0 Quick! n2. Discuss why as long as ! !n n2 x a cos !t has solution xp a cos !t !n 2 !2 sin !t x !n2 !n 2 !2 3. What about x !n2.

 

Gain and phase lag; resonance; undetermined coe cients Exponential response formula: A solution to p D x Aert is given by rt ep r 6ERF with resonance: Assume.

 

F requency Response.

 

Laplace transform Rules for the Laplace transform 1 De nition: L f t F s f t e st dt for Re s 0. 0 Linearity: L af t bg t aF s bG s. L 1: F s essen tially determines f t. s-shift rule: Lrtf t F s r. t-derivative: L f0 t sF s where f0 t denotes.

 

F ourier Series: Harmonic response If g x is a piecewise continuous periodic function and 2L is a period, then a0 x 2 x x 2 x g x a1 cos a2 cos b1 sin b2 sin 2 LL LL The F ourier coe cients are de ned as the numbers tting into this.

 

Convolution t Convolution product: f t g t f t g d 0 q t. 1. a Compute t u t. More generally, compute q t u t in terms of q t. b Compute u t.

 

Linear phase portraits The matrices I want you to study all have the form A a 2 2 1. 1. Compute the trace, determinant, c ha racteristic polynomial, and eigen v alues, in terms.

 

Exponential and sin usoidal input signals 2. For ! d4x 3. Find an exponential solution of x e 2t. dt4 d4x 4. Find a sinusoidal solution of.

 

Complex numbers, complex exponentials 1. Mark z 1 p3i on the co mplex plane. What is its polar co ordinates Then mark zn one in the form Aei Then mark zfor.

 

First order Linear ODEs: Integ rating factors Reduced standard form: x_ t p t x t q t , brie y x_ px q d In tegrating factor: a function u t such that.

 

Solutions to second order ODEs 1. Check that both x cos !t and x sin !t satisfy the second order linear di erential equation x !2 x 0 This equation is called.

 

18. 03 Class 36 , May 5,2010 The matrix exponential: initial value problems. 1. Definition of e At 2. Computation of e At 3. Uncoupled example 4. Defective example 5. Exponentiallaw 1 Recall from.

 

In particular: If all poles of F s have negative real part - are to the left of the imaginary axis - then f t --- 0 as t --- infinity. 2 This is a stability effect, related to one we have.