Background ODEs (2A) Young Won Lim 3/7/15
|
|
- Lindsay Stokes
- 5 years ago
- Views:
Transcription
1 Background ODEs (2A)
2 Copyright (c) Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using OpenOffice and Octave.
3 Second Order ODEs First Order Linear Equations a 1 (x) d y d x + a 0 (x) y = g(x) a 1 ( x) y ' + a 0 (x) y = g(x) Second Order Linear Equations a 2 (x) d 2 y d x 2 + a 1(x) d y d x + a 0(x) y = g(x) a 2 (x) y ' ' + a 1 (x) y ' + a 0 (x) y = g(x) Second Order Linear Equations with Constant Coefficients d 2 y a 2 d x + a d y 2 1 d x + a 0 y = g(x) a d2 y d x 2 + b d y d x + c y = g(x) a 2 y ' ' + a 1 y ' + a 0 y = g(x) a y ' ' + b y ' + c y = g(x) ODEs (BGND.2A) 3
4 Background ODEs (2A)
5 Second Order ODEs First Order Linear Equations a 1 (x) d y d x + a 0 (x) y = g(x) a 1 ( x) y ' + a 0 (x) y = g(x) Second Order Linear Equations a 2 (x) d 2 y d x 2 + a 1(x) d y d x + a 0(x) y = g(x) a 2 (x) y ' ' + a 1 (x) y ' + a 0 (x) y = g(x) Second Order Linear Equations with Constant Coefficients d 2 y a 2 d x + a d y 2 1 d x + a 0 y = g(x) a d2 y d x 2 + b d y d x + c y = g(x) a 2 y ' ' + a 1 y ' + a 0 y = g(x) a y ' ' + b y ' + c y = g(x) ODEs (BGND.2A) 5
6 Auxiliary Equation Homogeneous Second Order DEs with Constant Coefficients a d2 y d x 2 + b d y d x + c y = 0 a y ' ' + b y ' + c y = 0 try a solution y = e mx a d2 d x 2 {em x } + b d d x {em x } + c {e m x }= 0 a {m 2 e m x } + b{me m x } + c {e m x }= 0 (a m 2 + bm + c) e m x = 0 a {e m x }' ' + b{e m x }' + c {e m x }= 0 a {m 2 e m x } + b{me m x } + c {e m x }= 0 (a m 2 + b m + c) e m x = 0 auxiliary equation (a m 2 + b m + c) = 0 (a m 2 + bm + c) = 0 ODEs (BGND.2A) 6
7 Roots of the Auxiliary Equation Homogeneous Second Order DEs with Constant Coefficients a d2 y d x 2 + b d y d x + c y = 0 a y ' ' + b y ' + c y = 0 auxiliary equation y = e mx try a solution (a m 2 + bm + c) = 0 m 1 = ( b + b 2 4ac)/2a m 2 = ( b b 2 4ac)/2a = e m 1 x, = e m 2 x (A) b 2 4ac > 0 Real, distinct m 1, m 2 = e m 1 x = = e m 2 x (B) b 2 4ac = 0 Real, equal m 1, m 2 = e m 1 x, = e m 2 x (C) b 2 4ac < 0 Conjugate complex m 1, m 2 ODEs (BGND.2A) 7
8 Linear Combination of Solutions DEQ a d2 y d x 2 + b d y d x + c y = 0 a ' ' + b ' + c = 0 a ' ' + b ' + c = 0 C 1 + C 2 a( ' '+ ' ') + b( '+ ') + c( + ) = 0 a( + )' ' + b( + )' + c( + ) = 0 y 3 = + y 4 = a(c 1 ' '+C 2 ' ') + b(c 1 '+C 2 ') + c(c 1 +C 2 ) = 0 a(c 1 +C 2 )' ' + b(c 1 +C 2 )' + c(c 1 +C 2 ) = 0 y 5 = y y 4 y 6 = y 3 2 y 4 ODEs (BGND.2A) 8
9 Solutions of 2nd Order ODEs DEQ a d2 y d x 2 + b d y d x + c y = 0 m = e 1 x = e m 2 x = e m 1 x = e m 1 x = e m 2 x (D > 0) (D = 0) (D < 0) C 1 + C 2 y = C 1 e m1x + C 2 e m 2 x y = C 1 e m 1x? y = C 1 e m1x + C 2 e m 2 x (D > 0) (D = 0) (D < 0) auxiliary equation (a m 2 + bm + c) = 0 m 1 = ( b + b 2 4ac)/2a m 2 = ( b b 2 4ac)/2a ODEs (BGND.2A) 9
10 Fundamental Set of Solutions Second Order EQ a d2 y d x 2 + b d y d x + c y = 0 Functions and are either linearly independent functions or linearly dependent functions C 1 + C 2 {, } Second Order there can be at most two linearly independent functions y = C 1 e m 1x + C 2 e m 2 x y = C 1 e m 1x + C 2 x e m 1x any n linearly independent solutions of the homogeneous linear n-th order differential equation y = C 1 e m 1x + C 2 e m 2 x = C 1 e (α+i β) x + C 2 e (α iβ)x Fundamental Set of Solutions ODEs (BGND.2A) 10
11 (A) Real Distinct Roots Case Homogeneous Second Order DEs with Constant Coefficients a d2 y d x 2 + b d y d x + c y = 0 a y ' ' + b y ' + c y = 0 y = e mx try a solution (a m 2 + b m + c) = 0 auxiliary equation m 1 = ( b + b 2 4ac)/2 a m 2 = ( b b 2 4ac)/2a = e m 1 x = e m 2 x b 2 4ac > 0 b 2 4ac = 0 b 2 4ac < 0 Real, distinct m 1, m 2 Real, equal m 1, m 2 Conjugate complex m 1, m 2 y = C 1 e m 1x + C 2 e m 2 x y = C 1 e m 1x + C 2 x e m 1x y = C 1 e m 1x + C 2 e m 2 x = C 1 e (α+i β) x + C 2 e (α iβ)x ODEs (BGND.2A) 11
12 (B) Repeated Real Roots Case Homogeneous Second Order DEs with Constant Coefficients a d2 y d x 2 + b d y d x + c y = 0 a y ' ' + b y ' + c y = 0 y = e mx try a solution (a m 2 + bm + c) = 0 auxiliary equation m 1 = ( b + b 2 4ac)/2a m 2 = ( b b 2 4ac)/2a b 2 4ac = 0 m 1 = b/2 a m 2 = b /2 a e m 1 x = e m 2 x = e b 2 a x b 2 4ac > 0 b 2 4ac = 0 b 2 4ac < 0 Real, distinct m 1, m 2 Real, equal m 1, m 2 Conjugate complex m 1, m 2 y = C 1 e m 1x + C 2 e m 2 x y = C 1 e m 1x + C 2 x e m 1x y = C 1 e m 1x + C 2 e m 2 x = C 1 e (α+i β) x + C 2 e (α iβ)x ODEs (BGND.2A) 12
13 (C) Complex Roots of the Auxiliary Equation Homogeneous Second Order DEs with Constant Coefficients a d2 y d x 2 + b d y d x + c y = 0 a y ' ' + b y ' + c y = 0 y = e mx try a solution (a m 2 + bm + c) = 0 auxiliary equation m 1 = ( b + b 2 4 ac)/2a m 2 = ( b b 2 4a c)/2a m 1 = ( b + 4 ac b 2 i)/2a m 2 = ( b 4 a c b 2 i)/2a = e m 1 x = e m 2 x b 2 4ac > 0 b 2 4ac = 0 b 2 4ac < 0 Real, distinct m 1, m 2 Real, equal m 1, m 2 Conjugate complex m 1, m 2 y = C 1 e m 1x + C 2 e m 2 x y = C 1 e m 1x + C 2 x e m 1x y = C 1 e m 1x + C 2 e m 2 x = C 1 e (α+i β) x + C 2 e (α iβ)x ODEs (BGND.2A) 13
14 Complex Exponential Conversion Homogeneous Second Order DEs with Constant Coefficients a d2 y d x 2 + b d y d x + c y = 0 a y ' ' + b y ' + c y = 0 m 1 = ( b + b 2 4 ac)/2a m 1 = ( b + 4 a c b 2 i)/2 a = α + i β = e m 1 x m 2 = ( b b 2 4a c)/2a m 2 = ( b 4 a c b 2 i)/2a = α iβ = e m 2 x b 2 4ac < 0 Conjugate complex m 1, m 2 y = C 1 e m 1x + C 2 e m 2 x = C 1 e (α+i β) x + C 2 e (α iβ)x Pick two homogeneous solution = {e (α+i β)x + e (α iβ) x }/2 = {e (α+iβ) x e (α i β) x }/2i = e α x cos(β x) = e α x sin (β x) (C 1 =+1/2, C 2 =+1/2) (C 1 =+1/2i, C 2 = 1/2i) y = C 3 e α x cos(β x) + C 4 e α x sin(β x)= e α x (C 3 cos(β x) + C 4 sin (β x)) ODEs (BGND.2A) 14
15 Fundamental Set Examples (1) Second Order EQ a d2 y d x 2 + b d y d x + c y = 0 (α +iβ) x e (α i β) x e y 3 = 1 y y 2 2 {e (α+i β) x + e (α iβ)x }/2 = e α x cos(β x) y 4 = 1 2i 1 y 2i 2 {e (α+i β) x e (α i β) x }/2i = e α x sin (β x) ODEs (BGND.2A) 15
16 Fundamental Set Examples (2) Second Order EQ a d2 y d x 2 + b d y d x + c y = 0 = = y 3 +i y 4 y 3 i y 4 (α +iβ) x e (α i β) x e y 3 = e α x cos(β x) e α x [cos(β x) + i sin(β x)] y 4 = e α x sin(β x) e α x [cos(β x) i sin(β x)] ODEs (BGND.2A) 16
17 General Solution Examples Second Order EQ a d2 y d x 2 + b d y d x + c y = 0 linearly independent Fundamental Set of Solutions {, }={e (α+i β) x, e (α iβ)x } C 1 + C 2 C 1 e (α +i β) x (α i β) x + C 2 e General Solution linearly independent Fundamental Set of Solutions {y 3, y 4 }={e α x cos(β x), e α x sin (β x)} c 3 y 3 + c 3 y 4 c 3 e α x cos(β x) + c 4 e α x sin(β x) = e α x (c 3 cos(β x) + c 4 sin(β x)) General Solution ODEs (BGND.2A) 17
18 Finding a Particular Solution - Undetermined Coefficients ODEs (BGND.2A) 18
19 Particular Solutions DEQ a d2 y d x 2 + b d y d x + c y = g(x) When coefficients are constant y p particular solution by a conjecture (I) FORM Rule (II) Multiplication Rule And g(x) = A constant or A polynomial or An exponential function or A sine and cosine functions or Finite sum and products of the above functions k P(x) = a n x n + a n 1 x n a 1 x 1 + a 0 e α x sin(β x) cos(β x) e α x sin (β x) + x 2 And g(x) ln x 1 x tan x sin 1 x ODEs (BGND.2A) 19
20 Form Rule DEQ a d2 y d x 2 + b d y d x + c y = g(x) When coefficients are constant y p particular solution by a conjecture (I) FORM Rule (II) Multiplication Rule g(x) = 2 y p = A g(x) = 3 x+4 y p = Ax+B g(x) = 5 x 2 y p = Ax 2 +Bx+C g(x) = 6 x 2 7 y p = Ax 2 +Bx+C g(x) = sin 8 x y p = A cos 8 x+ B sin 8 x g(x) = cos 9 x y p = A cos 9 x+ B sin 9 x g(x) = e 10 x y p = A e 10 x g(x) = x e 11 x y p = ( Ax+ B)e 11 x g(x) = e 11 x sin 12 x y p = A e 11 x sin 12x+ B e 11 x cos12 x ODEs (BGND.2A) 20
21 Multiplication Rule DEQ a d2 y d x 2 + b d y d x + c y = g(x) y p use y p =x n y p =x n if y p = y p = Associated DEQ a d2 y d x 2 + b d y d x + c y = 0 y p + y c Any y p contains a term which is the same term in y c Use y p multiplied by x n n is the smallest positive integer that eliminates the duplication c 1 + c 2 ODEs (BGND.2A) 21
22 Example Form Rule (1) DEQ d 2 y d x + 3 d y 2 d x + 2 y = x y p + y p y c y p = Ax+B y p '= A y p ' '=0 y p ' ' + 3 y p ' + 2 y p = 3 A + 2(A x+b) = 2 A x+3 A+2 B = x Associated DEQ d 2 y d x d y d x + 2 y = 0 m 2 + 3m + 2 = 0 (m+2)(m+1) = 0 2 A = 1 3 A+2 B = 0 A = 1 2 B = 3 4 y p = 1 2 x 3 4 c 1 + c 2 y = c 1 e x + c 2 e 2x x 3 4 ODEs (BGND.2A) 22
23 Example Multiplication Rule (2) DEQ d 2 y d x 2 d y 2 d x + y = 2ex y p y ' ' 2 y '+ y = 2 e x y p = A e x A x e x A x 2 e x y p + y c Associated DEQ d 2 y d x 2 d y 2 = e x d x + y = 0 y ' ' 2 y '+ y = 0 = x e x c 1 + c 2 ODEs (BGND.2A) 23
24 Example Multiplication Rule (3) DEQ d 2 y d x 2 d y 2 d x + y = 2ex y p y ' ' 2 y '+ y = 6 x e x y p = A x e x A x 2 e x A x 3 e x y p + y c LHS :2 A e x Associated DEQ d 2 y d x 2 2 d y d x + y = 0 = e x y ' ' 2 y '+ y = 0 = x e x c 1 + c 2 ODEs (BGND.2A) 24
25 Three Differential Equations y p particular solutions EQ 1 d 2 y (t ) d t 2 + a 1 d y(t) d t + a 2 y(t) = x 1 ( x) + y h EQ 2 d 2 y (t ) d t 2 + a 1 d y(t) d t + a 2 y(t) = x 2 ( x) + y h d 2 y (t ) d t 2 + a 1 d y(t) d t + a 2 y(t) = 0 EQ 3 d 2 y (t ) d t 2 + a 1 d y(t) d t + a 2 y(t) = x 3 (x) y 3 + y h general solutions y h homogeneous solution ODEs (BGND.2A) 25
26 Superposition (1) DEQ d 2 y d x + b d y 2 d x + c y = 2 x cos8 x (2 x 2 + 3) + (cos8 x) d 2 y d x + b d y 2 d x + c y = 0 y c y c = c 1 e m 1x + c 2 e m 2x d 2 y d x + b d y 2 d x + c y = 2 x2 + 3 y p1 y p1 = A x 2 + B x + C d 2 y d x + b d y 2 d x + c y = cos 8 x y p2 y p2 = E cos 8 x + F sin 8 x d 2 d x 2 [ y c+ y p1 + y p2 ] + b d d x [ y c+ y p1 + y p2 ] + c[ y c + y p1 + y p2 ] = 2 x cos8 x ODEs (BGND.2A) 26
27 Superposition (2) DEQ a d2 y d x 2 + b d y d x + c y = (2 x2 + 3) cos 8 x d 2 y d x + b d y 2 d x + c y = 0 d 2 y d x + b d y 2 d x + c y = (2 x2 + 3) d 2 y d x + b d y 2 d x + c y = cos 8 x y c y p1 y p2 y p = ( A x 2 +B x+c) (cos 8 x+sin 8 x) d 2 d x 2 [ y c+ y p1 y p2 ] + b d d x [ y c+ y p1 y p2 ] + c [ y c + y p1 y p2 ] = (2 x 2 + 3) cos8 x ODEs (BGND.2A) 27
28 Finite Number of Derivative Functions y = x e m x y = 2 x x + 4 ẏ = e m x + m x e m x ẏ = 4 x + 3 ÿ = me m x + m(e m x + m x e m x ) = 2me m x + m 2 x e m x ÿ = 4 y = 2me mx + m 2 (e m x + m x e m x ) = (m 2 + 2m)e m x + m 3 x e m x y = 0 {e m x, x e m x } {2 x x + 4, 4 x + 3, 4} ODEs (BGND.2A) 28
29 Infinite Number of Derivative Functions y = +x 1 y = ln x ẏ = x 2 y = + x 1 ÿ = +2 x 3 ẏ = x 2 y = 6 x 4 ÿ = +2 x 3 y = 6 x 4 ODEs (BGND.2A) 29
30 Complex Exponentials c 1 e σ t +i ωt e + c 2 e σ t e iωt = e σ t (c 1 e +i ωt + c 2 e iω t ) (c 1 + c 2 ) = c 3 i(c 1 c 2 ) = c 4 = e σ t [c 1 (cos(ωt) + isin(ωt )) + c 2 (cos(ω t) i sin(ωt ))] = e σ t [(c 1 + c 2 )cos(ωt ) + i(c 1 c 2 ) sin(ω t)] = c 3 e σt cos(ωt ) + c 4 e σ t sin(ω t) c 3 e σt cos(ωt ) + c 4 e σ t sin(ω t) = c 3 e σ t (e +i ωt +e iωt )/2 + c 4 e σ t (e +iω t e iω t )/2i (c 3 c 4 i) 2 (c 3 +c 4 i) 2 = c 1 = c 2 = c 3 e σ t (e +i ωt +e iωt )/2 + c 4 e σ t ( i e +iω t +i e iωt )/2 = (c 3 c 4 i) 2 e σ t e +iω t + (c 3+c 4 i) 2 = c 1 e σt e +iωt + c 2 e σ t e iω t e σ t i ωt e ODEs (BGND.2A) 30
31 Complex Exponentials c 1 e σ t +i ωt e + c 2 e σ t e iωt c 3 e σt cos(ωt ) + c 4 e σ t sin(ω t) (c 1 + c 2 ) = c 3 i(c 1 c 2 ) = c 4 c 1 e σ t +i ωt e + c 2 e σ t e iωt c 3 e σt cos(ωt ) + c 4 e σ t sin(ω t) c 1 = (c 3 c 4 i) 2 c 2 = (c 3+c 4 i) 2 ODEs (BGND.2A) 31
32 Basis of the Complex Plane Basis : a set of linear independent spanning vectors every complex number can be represented by e + j θ e j θ k 1 e + j θ + k 2 e + j θ linear combination of e + j θ and e + j θ which are one set of linear independent two vectors every complex number can also be represented by l 1 cosθ + l 2 j sin θ j sin θ e + j θ j sinθ l 1 cosθ + l 2 j sinθ cosθ cosθ e j θ ODEs (BGND.2A) 32
33 Real Coefficients C 1 & C 2 c 1 e +iω t i ωt + c 2 e c 3 cos(ω t) + c 4 sin(ω t) real number real number c 1 c 2 = (c 3 c 4 i)/2 = (c 3 +c 4 i)/2 c 3 = (c 1 + c 2 ) c 4 = i(c 1 c 2 ) real number imaginary number (c 1 e σ t, c 2 e σ t ) +i ωt 1 e + 0 e iωt 1 2 e+i ωt + 1 ωt e i 2 e i ω t e +i ω t 1 cos(ωt) + 1i sin(ω t) 2 cos(ωt ) + 0i sin(ω t) 0 e +iωt + 1 e iωt 1 cos(ωt) 1i e σt sin(ωt ) 1 2 e+i ωt + 1 ωt e i 2 0 cos(ω t) 2i sin(ωt) 1 e +iω t + 0 e iωt isin (ω t ) 1 cos(ω t) 1i sin(ω t) 1 ωt e+i e +iωt 1 ωt e+i e iωt 1 e iω t 1 ωt e i 2 cos(ω t) 2 cos(ωt ) 0i sin(ω t) 1 cos(ω t) + 1i sin(ω t) 0 cos(ω t) + 2i sin(ω t) (c 3 e σ t, c 4 e σ t ) ODEs (BGND.2A) 33
34 Complex Plane Basis c 1 e +iω t i ωt + c 2 e c 3 ' cos(ωt) + c 4 ' isin (ω t) c 1 c 2 = (c 3 '+c 4 ')/2 = (c 3 ' c 4 ')/2 real number real number C 1 c 3 ' = (c 1 + c 2 ) c 4 ' = (c 1 c 2 ) real number real number +i ωt 1 e + 0 e iωt 1 2 e+i ωt + 1 ωt e i 2 0 e +iωt + 1 e iωt (c 1, c 2 ) e i ω t e +i ω t 1 cos(ωt) + 1i sin(ω t) 2 cos(ωt ) + 0i sin(ω t) 1 cos(ωt) 1i sin(ωt) 1 2 e+i ωt + 1 ωt e i 2 1 e +iω t + 0 e iωt (c 3, c 4 ) isin (ω t ) complex plane 0 cos(ω t) 2i sin(ωt) 1 cos(ω t) 1i sin(ω t) 1 ωt e+i e +iωt 1 ωt e+i e iωt 1 e iω t 1 ωt e i 2 cos(ω t) 2 cos(ωt ) 0i sin(ω t) 1 cos(ω t) + 1i sin(ω t) 0 cos(ω t) + 2i sin(ω t) ODEs (BGND.2A) 34
35 Real Coefficients C 3 & C 4 c 1 e +iω t i ωt + c 2 e c 3 cos(ω t) + c 4 sin(ω t) c 1 c 2 = (c 3 c 4 i)/2 = (c 3 +c 4 i)/2 conjugate complex number R 2 +2 real part 2 imag part c 3 = (c 1 + c 2 ) c 4 = i(c 1 c 2 ) real number real number (+1 0i) 2 e +iω t + (+1+0 i) 2 e iωt 1 cos(ωt) + 0 sin(ω t) (+ 1 i) 2 2 e+iω t + (+1+i) 2 2 e iω t (0 i) 2 e+i ωt + (0+i) ωt 2 e i 1 cos(ωt) + 1 sin(ωt ) cos(ω t) + 1 sin(ω t) ( 1 i) ωt e+i ( 1+i) ωt e i 2 2 ( 1 0 i) 2 e +i ωt + ( 1+0i) i ωt 2 e real plane 1 cos(ωt ) + 1 sin (ω t) cos(ω t) + 0 sin(ω t) ( 1+i) 2 2 e+i ωt + ( 1 i) ωt e i 2 2 (0+i) 2 e+i ωt + (0 i) ωt 2 e i (c 3, c 4 ) sin(ω t) cos(ωt ) + sin(ω t) 2 0 cos(ω t) 1 sin(ωt) (+ 1+i) ωt e+i (+1 i) 2 2 e iω t cos(ω t) 1 cos(ωt) 1 sin(ωt ) 2 2 ODEs (BGND.2A) 35
36 Real Coefficients C 3 & C 4 c 1 e +iω t i ωt + c 2 e c 3 cos(ω t) + c 4 sin(ω t) c 1 c 2 = (c 3 c 4 i)/2 = (c 3 +c 4 i)/2 conjugate complex number R 2 +2 real part 2 imag part c 3 = (c 1 + c 2 ) c 4 = i(c 1 c 2 ) real number real number A cos(ωt ϕ) c 3 cos(ω t) + c 4 sin(ω t) c 3 2 +c 4 2 c 3 c c 4 c 4 c c 4 = A = cos(ϕ) = sin(ϕ) ODEs (BGND.2A) 36
37 C 1 and R 2 Spaces c 1 e +iω t i ωt + c 2 e c 3 cos(ω t) + c 4 i sin(ωt ) c 1 = (c 3 +c 4 )/2 c 2 = (c 3 c 4 )/2 real number real number c 3 = (c 1 + c 2 ) c 4 = (c 1 c 2 ) real number real number C 1 isin (ω t ) cos(ω t) c 1 e +iω t i ωt + c 2 e c 3 cos(ω t) + c 4 sin(ω t) c 1 = (c 3 c 4 i)/2 conjugate c 2 = (c 3 + c 4 i)/2 complex number +2 real part 2 imag part c 3 = (c 1 + c 2 ) c 4 = i(c 1 c 2 ) real number real number R 2 sin(ω t) cos(ω t) ODEs (BGND.2A) 37
38 Signal Spaces and Phasors c 1 e +iω t i ωt + c 2 e c 3 cos(ω t) + c 4 sin(ω t) c 1 = (c 3 c 4 i)/2 conjugate c 2 = (c 3 + c 4 i)/2 complex number +2 real part 2 imag part c 3 = (c 1 + c 2 ) c 4 = i(c 1 c 2 ) real number real number R 2 sin(ω t) cos(ω t) ODEs (BGND.2A) 38
39 References [1] [2] M.L. Boas, Mathematical Methods in the Physical Sciences [3] E. Kreyszig, Advanced Engineering Mathematics [4] D. G. Zill, W. S. Wright, Advanced Engineering Mathematics [5]
CLTI Differential Equations (3A) Young Won Lim 6/4/15
CLTI Differential Equations (3A) Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationSecond Order ODE's (2A) Young Won Lim 5/5/15
Second Order ODE's (2A) Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationLinear Equations with Constant Coefficients (2A) Young Won Lim 4/13/15
Linear Equations with Constant Coefficients (2A) Copyright (c) 2011-2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationPhasor Young Won Lim 05/19/2015
Phasor Copyright (c) 2009-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
More informationBackground Trigonmetry (2A) Young Won Lim 5/5/15
Background Trigonmetry (A) Copyright (c) 014 015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. or
More informationHigher Order ODE's (3A) Young Won Lim 12/27/15
Higher Order ODE's (3A) Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationHigher Order ODE's (3A) Young Won Lim 7/7/14
Higher Order ODE's (3A) Copyright (c) 2011-2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationBackground Complex Analysis (1A) Young Won Lim 9/2/14
Background Complex Analsis (1A) Copright (c) 2014 Young W. Lim. Permission is granted to cop, distribute and/or modif this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationHigher Order ODE's (3A) Young Won Lim 7/8/14
Higher Order ODE's (3A) Copyright (c) 2011-2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationGeneral Vector Space (2A) Young Won Lim 11/4/12
General (2A Copyright (c 2012 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
More informationHigher Order ODE's, (3A)
Higher Order ODE's, (3A) Initial Value Problems, and Boundary Value Problems Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms
More informationMatrix Transformation (2A) Young Won Lim 11/9/12
Matrix (A Copyright (c 01 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. or any later version published
More informationSeparable Equations (1A) Young Won Lim 3/24/15
Separable Equations (1A) Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationCT Rectangular Function Pairs (5B)
C Rectangular Function Pairs (5B) Continuous ime Rect Function Pairs Copyright (c) 009-013 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU
More informationIntroduction to ODE's (0P) Young Won Lim 12/27/14
Introuction to ODE's (0P) Copyright (c) 2011-2014 Young W. Lim. Permission is grante to copy, istribute an/or moify this ocument uner the terms of the GNU Free Documentation License, Version 1.2 or any
More informationGeneral CORDIC Description (1A)
General CORDIC Description (1A) Copyright (c) 2010, 2011, 2012 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License,
More informationIntroduction to ODE's (0A) Young Won Lim 3/9/15
Introduction to ODE's (0A) Copyright (c) 2011-2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More informationDefinitions of the Laplace Transform (1A) Young Won Lim 1/31/15
Definitions of the Laplace Transform (A) Copyright (c) 24 Young W. Lim. Permission is granted to copy, distriute and/or modify this document under the terms of the GNU Free Documentation License, Version.2
More informationODE Background: Differential (1A) Young Won Lim 12/29/15
ODE Background: Differential (1A Copyright (c 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationHilbert Inner Product Space (2B) Young Won Lim 2/7/12
Hilbert nner Product Space (2B) Copyright (c) 2009-2011 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationLine Integrals (4A) Line Integral Path Independence. Young Won Lim 10/22/12
Line Integrals (4A Line Integral Path Independence Copyright (c 2012 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License,
More informationComplex Series (3A) Young Won Lim 8/17/13
Complex Series (3A) 8/7/3 Copyright (c) 202, 203 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or
More informationMatrix Transformation (2A) Young Won Lim 11/10/12
Matrix (A Copyright (c 0 Young W. im. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation icense, Version. or any later version published
More informationdx n a 1(x) dy
HIGHER ORDER DIFFERENTIAL EQUATIONS Theory of linear equations Initial-value and boundary-value problem nth-order initial value problem is Solve: a n (x) dn y dx n + a n 1(x) dn 1 y dx n 1 +... + a 1(x)
More informationCapacitor and Inductor
Capacitor and Inductor Copyright (c) 2015 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationCapacitor and Inductor
Capacitor and Inductor Copyright (c) 2015 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationFourier Analysis Overview (0A)
CTFS: Fourier Series CTFT: Fourier Transform DTFS: Fourier Series DTFT: Fourier Transform DFT: Discrete Fourier Transform Copyright (c) 2011-2016 Young W. Lim. Permission is granted to copy, distribute
More informationDifferentiation Rules (2A) Young Won Lim 1/30/16
Differentiation Rules (2A) Copyright (c) 2011-2016 Young W. Lim. Permission is grante to copy, istribute an/or moify this ocument uner the terms of the GNU Free Documentation License, Version 1.2 or any
More informationComplex Trigonometric and Hyperbolic Functions (7A)
Complex Trigonometric and Hyperbolic Functions (7A) 07/08/015 Copyright (c) 011-015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationComplex Functions (1A) Young Won Lim 2/22/14
Complex Functions (1A) Copyright (c) 2011-2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationBackground LTI Systems (4A) Young Won Lim 4/20/15
Background LTI Systems (4A) Copyright (c) 2014-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More informationFourier Analysis Overview (0A)
CTFS: Fourier Series CTFT: Fourier Transform DTFS: Fourier Series DTFT: Fourier Transform DFT: Discrete Fourier Transform Copyright (c) 2011-2016 Young W. Lim. Permission is granted to copy, distribute
More informationGroup & Phase Velocities (2A)
(2A) 1-D Copyright (c) 2011 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published
More informationDigital Signal Octave Codes (0A)
Digital Signal Periodic Conditions Copyright (c) 2009-2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationDispersion (3A) 1-D Dispersion. Young W. Lim 10/15/13
1-D Dispersion Copyright (c) 013. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. or any later version published
More informationDigital Signal Octave Codes (0A)
Digital Signal Periodic Conditions Copyright (c) 2009-2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationExpected Value (10D) Young Won Lim 6/12/17
Expected Value (10D) Copyright (c) 2017 Young W. Lim. Permissios granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
More informationDigital Signal Octave Codes (0A)
Digital Signal Periodic Conditions Copyright (c) 2009-207 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2
More informationDifferentiation Rules (2A) Young Won Lim 2/22/16
Differentiation Rules (2A) Copyright (c) 2011-2016 Young W. Lim. Permission is grante to copy, istribute an/or moify this ocument uner the terms of the GNU Free Documentation License, Version 1.2 or any
More informationDigital Signal Octave Codes (0A)
Digital Signal Periodic Conditions Copyright (c) 2009-207 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2
More informationPropagating Wave (1B)
Wave (1B) 3-D Wave Copyright (c) 2011 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
More informationDigital Signal Octave Codes (0A)
Digital Signal Periodic Conditions Copyright (c) 2009-207 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2
More informationCLTI System Response (4A) Young Won Lim 4/11/15
CLTI System Response (4A) Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More informationLine Integrals (4A) Line Integral Path Independence. Young Won Lim 11/2/12
Line Integrals (4A Line Integral Path Independence Copyright (c 2012 Young W. Lim. Permission is granted to copy, distriute and/or modify this document under the terms of the GNU Free Documentation License,
More informationSection 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More informationRelations (3A) Young Won Lim 3/27/18
Relations (3A) Copyright (c) 2015 2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
More information3.4.1 Distinct Real Roots
Math 334 3.4. CONSTANT COEFFICIENT EQUATIONS 34 Assume that P(x) = p (const.) and Q(x) = q (const.), so that we have y + py + qy = 0, or L[y] = 0 (where L := d2 dx 2 + p d + q). (3.7) dx Guess a solution
More informationSolution of Constant Coefficients ODE
Solution of Constant Coefficients ODE Department of Mathematics IIT Guwahati Thus, k r k (erx ) r=r1 = x k e r 1x will be a solution to L(y) = 0 for k = 0, 1,..., m 1. So, m distinct solutions are e r
More informationThe Growth of Functions (2A) Young Won Lim 4/6/18
Copyright (c) 2015-2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published
More informationSignal Functions (0B)
Signal Functions (0B) Signal Functions Copyright (c) 2009-203 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License,
More informationDigital Signal Octave Codes (0A)
Digital Signal Periodic Conditions Copyright (c) 2009-207 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2
More informationDFT Frequency (9A) Each Row of the DFT Matrix. Young Won Lim 7/31/10
DFT Frequency (9A) Each ow of the DFT Matrix Copyright (c) 2009, 2010 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GU Free Documentation License,
More informationRoot Locus (2A) Young Won Lim 10/15/14
Root Locus (2A Copyright (c 2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
More informationSpectra (2A) Young Won Lim 11/8/12
Spectra (A) /8/ Copyright (c) 0 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version. or any later version
More informationGeneral Vector Space (3A) Young Won Lim 11/19/12
General (3A) /9/2 Copyright (c) 22 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later version
More informationSignals and Spectra (1A) Young Won Lim 11/26/12
Signals and Spectra (A) Copyright (c) 202 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later
More informationFourier Analysis Overview (0B)
CTFS: Continuous Time Fourier Series CTFT: Continuous Time Fourier Transform DTFS: Fourier Series DTFT: Fourier Transform DFT: Discrete Fourier Transform Copyright (c) 2009-2016 Young W. Lim. Permission
More informationSection 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation. is called the reduced equation of (N).
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More informationChapter 2: Complex numbers
Chapter 2: Complex numbers Complex numbers are commonplace in physics and engineering. In particular, complex numbers enable us to simplify equations and/or more easily find solutions to equations. We
More informationCapacitor in an AC circuit
Capacitor in an AC circuit Copyright (c) 2011 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More informationDiscrete Time Rect Function(4B)
Discrete Time Rect Function(4B) Discrete Time Rect Functions Copyright (c) 29-213 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationUNDETERMINED COEFFICIENTS SUPERPOSITION APPROACH *
4.4 UNDETERMINED COEFFICIENTS SUPERPOSITION APPROACH 19 Discussion Problems 59. Two roots of a cubic auxiliary equation with real coeffi cients are m 1 1 and m i. What is the corresponding homogeneous
More informationGroup Velocity and Phase Velocity (1A) Young Won Lim 5/26/12
Group Velocity and Phase Velocity (1A) Copyright (c) 211 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationDiscrete Time Rect Function(4B)
Discrete Time Rect Function(4B) Discrete Time Rect Functions Copyright (c) 29-23 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationSurface Integrals (6A)
Surface Integrals (6A) Surface Integral Stokes' Theorem Copright (c) 2012 Young W. Lim. Permission is granted to cop, distribute and/or modif this document under the terms of the GNU Free Documentation
More informationHyperbolic Functions (1A)
Hyperbolic Functions (A) 08/3/04 Copyright (c) 0-04 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version. or
More informationGroup Delay and Phase Delay (1A) Young Won Lim 7/19/12
Group Delay and Phase Delay (A) 7/9/2 Copyright (c) 2 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2
More informationHigher-order ordinary differential equations
Higher-order ordinary differential equations 1 A linear ODE of general order n has the form a n (x) dn y dx n +a n 1(x) dn 1 y dx n 1 + +a 1(x) dy dx +a 0(x)y = f(x). If f(x) = 0 then the equation is called
More informationDetect Sensor (6B) Eddy Current Sensor. Young Won Lim 11/19/09
Detect Sensor (6B) Eddy Current Sensor Copyright (c) 2009 Young W. Lim. Permission is granteo copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationOrdinary Differential Equations
Ordinary Differential Equations (MA102 Mathematics II) Shyamashree Upadhyay IIT Guwahati Shyamashree Upadhyay ( IIT Guwahati ) Ordinary Differential Equations 1 / 15 Method of Undetermined Coefficients
More informationMath 211. Substitute Lecture. November 20, 2000
1 Math 211 Substitute Lecture November 20, 2000 2 Solutions to y + py + qy =0. Look for exponential solutions y(t) =e λt. Characteristic equation: λ 2 + pλ + q =0. Characteristic polynomial: λ 2 + pλ +
More informationCapacitor Young Won Lim 06/22/2017
Capacitor Copyright (c) 2011 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
More informationMath 4B Notes. Written by Victoria Kala SH 6432u Office Hours: T 12:45 1:45pm Last updated 7/24/2016
Math 4B Notes Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: T 2:45 :45pm Last updated 7/24/206 Classification of Differential Equations The order of a differential equation is the
More informationAudio Signal Generation. Young Won Lim 2/2/18
Generation Copyright (c) 2016-2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
More informationTest #2 Math 2250 Summer 2003
Test #2 Math 225 Summer 23 Name: Score: There are six problems on the front and back of the pages. Each subpart is worth 5 points. Show all of your work where appropriate for full credit. ) Show the following
More informationMath 2Z03 - Tutorial # 6. Oct. 26th, 27th, 28th, 2015
Math 2Z03 - Tutorial # 6 Oct. 26th, 27th, 28th, 2015 Tutorial Info: Tutorial Website: http://ms.mcmaster.ca/ dedieula/2z03.html Office Hours: Mondays 3pm - 5pm (in the Math Help Centre) Tutorial #6: 3.4
More informationB Ordinary Differential Equations Review
B Ordinary Differential Equations Review The profound study of nature is the most fertile source of mathematical discoveries. - Joseph Fourier (1768-1830) B.1 First Order Differential Equations Before
More informationBayes Theorem (10B) Young Won Lim 6/3/17
Bayes Theorem (10B) Copyright (c) 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
More informationComplex Numbers. The set of complex numbers can be defined as the set of pairs of real numbers, {(x, y)}, with two operations: (i) addition,
Complex Numbers Complex Algebra The set of complex numbers can be defined as the set of pairs of real numbers, {(x, y)}, with two operations: (i) addition, and (ii) complex multiplication, (x 1, y 1 )
More informationSection 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation. is called the reduced equation of (N).
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More informationOrdinary Differential Equations (ODEs)
Chapter 13 Ordinary Differential Equations (ODEs) We briefly review how to solve some of the most standard ODEs. 13.1 First Order Equations 13.1.1 Separable Equations A first-order ordinary differential
More informationUp-Sampling (5B) Young Won Lim 11/15/12
Up-Sampling (5B) Copyright (c) 9,, Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version. or any later version
More informationSection 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation. is called the reduced equation of (N).
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More informationMagnetic Sensor (3B) Magnetism Hall Effect AMR Effect GMR Effect. Young Won Lim 9/23/09
Magnetic Sensor (3B) Magnetism Hall Effect AMR Effect GMR Effect Copyright (c) 2009 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationDown-Sampling (4B) Young Won Lim 10/25/12
Down-Sampling (4B) /5/ Copyright (c) 9,, Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version. or any later
More informationCapacitor in an AC circuit
Capacitor in an AC circuit Copyright (c) 2011 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More informationAudio Signal Generation. Young Won Lim 1/29/18
Generation Copyright (c) 2016-2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
More informationSolutions to the Homework Replaces Section 3.7, 3.8
Solutions to the Homework Replaces Section 3.7, 3.8. Show that the period of motion of an undamped vibration of a mass hanging from a vertical spring is 2π L/g SOLUTION: With no damping, mu + ku = 0 has
More informationCapacitor in an AC circuit
Capacitor in an AC circuit Copyright (c) 2011 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More informationNotes on the Periodically Forced Harmonic Oscillator
Notes on the Periodically orced Harmonic Oscillator Warren Weckesser Math 38 - Differential Equations 1 The Periodically orced Harmonic Oscillator. By periodically forced harmonic oscillator, we mean the
More informationExamples: Solving nth Order Equations
Atoms L. Euler s Theorem The Atom List First Order. Solve 2y + 5y = 0. Examples: Solving nth Order Equations Second Order. Solve y + 2y + y = 0, y + 3y + 2y = 0 and y + 2y + 5y = 0. Third Order. Solve
More informationConsider an ideal pendulum as shown below. l θ is the angular acceleration θ is the angular velocity
1 Second Order Ordinary Differential Equations 1.1 The harmonic oscillator Consider an ideal pendulum as shown below. θ l Fr mg l θ is the angular acceleration θ is the angular velocity A point mass m
More informationDifferential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Undetermined Coefficients Page 1
Differential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Undetermined Coefficients Page 1 Questions Example (3.5.3) Find a general solution of the differential equation y 2y 3y = 3te
More informationSolutions to the Homework Replaces Section 3.7, 3.8
Solutions to the Homework Replaces Section 3.7, 3.8 1. Our text (p. 198) states that µ ω 0 = ( 1 γ2 4km ) 1/2 1 1 2 γ 2 4km How was this approximation made? (Hint: Linearize 1 x) SOLUTION: We linearize
More informationCapacitor in an AC circuit
Capacitor in an AC circuit Copyright (c) 2011 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More informationCapacitor in an AC circuit
Capacitor in an AC circuit Copyright (c) 2011 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More informationChapter 10: Sinusoidal Steady-State Analysis
Chapter 10: Sinusoidal Steady-State Analysis 1 Objectives : sinusoidal functions Impedance use phasors to determine the forced response of a circuit subjected to sinusoidal excitation Apply techniques
More informationDefinitions of the Laplace Transform (1A) Young Won Lim 2/9/15
Definition of the aplace Tranform (A) 2/9/5 Copyright (c) 24 Young W. im. Permiion i granted to copy, ditriute and/or modify thi document under the term of the GNU Free Documentation icene, Verion.2 or
More informationMAE 200B Homework #3 Solutions University of California, Irvine Winter 2005
Problem 1 (Haberman 5.3.2): Consider this equation: MAE 200B Homework #3 Solutions University of California, Irvine Winter 2005 a) ρ 2 u t = T 2 u u 2 0 + αu + β x2 t The term αu describes a force that
More informationDown-Sampling (4B) Young Won Lim 11/15/12
Down-Sampling (B) /5/ Copyright (c) 9,, Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version. or any later
More information