Background ODEs (2A)
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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
Background ODEs (2A)
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
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
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
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 3 + 2 y 4 y 6 = y 3 2 y 4 ODEs (BGND.2A) 8
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
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
(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
(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
(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
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
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 2 1 + 1 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
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
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
Finding a Particular Solution - Undetermined Coefficients ODEs (BGND.2A) 18
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 1 + + 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
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
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
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 2 + 3 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 + 1 2 x 3 4 ODEs (BGND.2A) 22
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
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
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
Superposition (1) DEQ d 2 y d x + b d y 2 d x + c y = 2 x2 + 3 + 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 2 + 3 + cos8 x ODEs (BGND.2A) 26
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
Finite Number of Derivative Functions y = x e m x y = 2 x 2 + 3 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 2 + 3 x + 4, 4 x + 3, 4} ODEs (BGND.2A) 28
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
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
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
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
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 + 2 0 e +iωt 1 ωt e+i + 2 1 2 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
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 + 2 0 e +iωt 1 ωt e+i + 2 1 2 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
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 ) 2 2 0 cos(ω t) + 1 sin(ω t) ( 1 i) ωt e+i + 2 2 ( 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) 2 2 1 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) 1 2 1 cos(ωt ) + sin(ω t) 2 0 cos(ω t) 1 sin(ωt) (+ 1+i) ωt e+i + 2 2 (+1 i) 2 2 e iω t cos(ω t) 1 cos(ωt) 1 sin(ωt ) 2 2 ODEs (BGND.2A) 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 A cos(ωt ϕ) c 3 cos(ω t) + c 4 sin(ω t) c 3 2 +c 4 2 c 3 c 2 2 3 +c 4 c 4 c 2 2 3 +c 4 = A = cos(ϕ) = sin(ϕ) ODEs (BGND.2A) 36
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
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
References [1] http://en.wikipedia.org/ [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] www.chem.arizona.edu/~salzmanr/480a