Linear Equations with Constant Coefficients (2A)
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Homogeneous Linear Equations with constant coefficients Linear Equations (1A) 3
Types of First Order ODEs A General Form of First Order Differential Equations d y d x = g(x, y) y ' = g(x, y) Separable Equations d y d x = g 1 (x)g 2 ( y) y ' = g 1 (x)g 2 ( y) y = f (x) 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) y = f (x) Exact Equations M (x, y)dx + N (x, y)dy = 0 z x d x + z y dy=0 z = f (x, y) Linear Equations (1A) 4
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) 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) Linear Equations (1A) 5
Auxiliary Equation Homogeneous Second Order DEs with Constant Coefficients 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 Linear Equations (1A) 6
Roots of the Auxiliary Equation Homogeneous Second Order DEs with Constant Coefficients 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 = e m 1 x = = e m 2 x (B) b 2 4ac = 0 Real, equal m 1 = e m 1 x, = e m 2 x (C) b 2 4ac < 0 Conjugate complex m 1 Linear Equations (1A) 7
Linear Combination of Solutions DEQ 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 Linear Equations (1A) 8
Solutions of 2nd Order ODEs DEQ 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 e m1x + C 2 e m 2 x e m 1x? 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 Linear Equations (1A) 9
Linear Independent Functions C 1 + C 2 = 0 C 1 = C 2 = 0 always zero means all coefficients must be zero and.are linearly independent functions if not all zero, can be represented by, and vice versa. C 1 0 = C 2 C 1 = a and.are linearly dependent functions simply a constant multiple of, and vice versa. C 2 0 = C 1 C 2 = b Linear Equations (1A) 10
Linear Independent Functions Example (1) C 1 C 2 many solutions of C1, C2 = e 2 x = 3 e 2 x = e2 x 3 e 2 x = 1 3 = c 3 {e 2 x } 1 {3e 2 x }= 0 6 {e 2 x }+2 {3 e 2x }= 0 linearly independent functions C 1 C 2 the only solution C1, C2 = e 2x = x e 2 x 0{e 2x }+0 {x e 2x } = 0 = e2 x x e 2 x = 1 x = u(x) linearly independent functions Linear Equations (1A) 11
Linear Independent Functions Example (2) linearly dependent functions = e 2 x = 3e 2 x y ' ' 4 y ' + 4 y = 0 m 2 4 m + 4 = 0 y = c 1 e 2 x + c 2 3 e 2 x y = (c 1 + 3 c 2 )e 2 x = C e 2 x linearly independent functions general solution = e 2x ' = 2 e 2 x ' ' = 4 e 2 x 4 = 4 e 2x 4 ' = 8 e 2 x ' ' = 4 e 2 x (m 2)2 = 0 = e 2x = x e 2 x 0 y = c 1 e 2 x + c 2 x e 2 x general solution = x e 2x ' = e 2 x + 2 x e 2 x ' ' = 2e 2x + 2e 2 x + 4 x e 2 x 4 = 4 x e 2 x 4 ' = 4 e 2 x 8 x e 2 x ' ' = 4 e 2 x + 4 x e 2 x 0 Linear Equations (1A) 12
Fundamental Set of Solutions Second Order EQ 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 e m 2 x x e m 1x any n linearly independent solutions of the homogeneous linear n-th order differential equation e m 2 x = C 1 e (α+i β) x + C 2 e (α iβ)x Fundamental Set of Solutions Linear Equations (1A) 13
Linear Independent Functions and Wronskian C 1 + C 2 = 0 C 1 = C 2 = 0 always zero means all coefficients must be zero and.are linearly independent functions C 1 + C 2 = 0 C 1 ' + C 2 ' = 0 [ ' '][ C 1 2] C = [ 0 0] If the inverse matrix exists ' ' 0 [ C 1 C 2] = [ ' '] 1[ 0 0] = [ 0 0] the only solution: trivial W (, ) 0 Linear Equations (1A) 14
(A) Real Distinct Roots Case Homogeneous Second Order DEs with Constant Coefficients 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 Real, equal m 1 Conjugate complex m 1 e m 2 x x e m 1x e m 2 x = C 1 e (α+i β) x + C 2 e (α iβ)x Linear Equations (1A) 15
(B) Repeated Real Roots Case Homogeneous Second Order DEs with Constant Coefficients 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 Real, equal m 1 Conjugate complex m 1 e m 2 x x e m 1x e m 2 x = C 1 e (α+i β) x + C 2 e (α iβ)x Linear Equations (1A) 16
(C) Complex Roots of the Auxiliary Equation Homogeneous Second Order DEs with Constant Coefficients 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 Real, equal m 1 Conjugate complex m 1 e m 2 x x e m 1x e m 2 x = C 1 e (α+i β) x + C 2 e (α iβ)x Linear Equations (1A) 17
Complex Exponential Conversion Homogeneous Second Order DEs with Constant Coefficients 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 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)) Linear Equations (1A) 18
Wronskian Second Order EQ a d 2 y d x + c y = 0 (α +iβ) x e (α i β) x e e (α+i β)x e (α iβ)x (e (α +iβ)x )' (e (α iβ)x )' = e (α+iβ)x e (α iβ)x (α+iβ)e (α+i β)x ' (α iβ)e (α i β) x = (α iβ)e 2 α x (α+iβ)e 2 α x = ( i2β)e 2α x 0 y 3 = 1 2 + 1 y 2 2 {e (α+i β) x + e (α i β) x }/2 = e α x cos(β x) y 4 = 1 2i 1 y 2i 2 {e (α+i β) x e (α iβ) x }/2 i = e α x sin(β x) e α x cos(β x) e α x sin(β x) (e α x cos(β x))' (e α x sin(β x))' = e α x cos(β x) e α x sin(β x) (α e α x cos(β x) β e α x sin(β x)) (α e α x sin(β x) + βe α x cos(β x)) = e α x cos(β x)(α e α x sin(β x) + βe α x cos(β x)) e α x sin(β x)(α e α x cos(β x) β e α x sin (β x)) = e α x cos(β x)β e α x cos(β x) + e α x sin (β x)βe α x sin(β x) = β e 2 α x (cos 2 (β x) + sin 2 (β x)) = β e 2 α x 0 Linear Equations (1A) 19
Wronskian : Linear Independent Second Order EQ a d 2 y d x + c y = 0 (α +iβ) x e (α i β) x e e (α+i β)x e (α iβ)x (e (α +iβ)x )' (e (α iβ)x )' 0 linearly independent Fundamental Set of Solutions y 3 = 1 2 + 1 y 2 2 {e (α+i β) x + e (α i β) x }/2 = e α x cos(β x) y 4 = 1 2i 1 y 2i 2 {e (α+i β) x e (α iβ) x }/2 i = e α x sin(β x) e α x cos(β x) e α x sin(β x) (e α x cos(β x))' (e α x sin(β x))' 0 linearly independent Fundamental Set of Solutions Linear Equations (1A) 20
Fundamental Set Examples (1) Second Order EQ 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) Linear Equations (1A) 21
Fundamental Set Examples (2) Second Order EQ 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)] Linear Equations (1A) 22
General Solution Examples Second Order EQ 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 Linear Equations (1A) 23
General Solutions - Homogeneous Equation - Non-homogeneous Equation Linear Equations (1A) 24
General Solution Homogeneous Equations Homogeneous Second Order DEs with Constant Coefficients d x + c y = 0 a y ' ' + b y ' + c y = 0 auxiliary equation (a m 2 + bm + c) = 0 m 1 = ( b + b 2 4ac)/2 a m 2 = ( b b 2 4ac)/2a (A) b 2 4ac > 0 Real, distinct m 1 e m 2 x (B) b 2 4ac = 0 Real, equal m 1 x e m 1x (C) b 2 4ac < 0 Conjugate complex m 1 e m 2 x = C 1 e (α+i β) x + C 2 e (α iβ)x Linear Equations (1A) 25
General Solution Nonhomogeneous Equations Nonhomogeneous Second Order DEs with Constant Coefficients d x + c y = g(x) a y ' ' + b y ' + c y = g(x) y = c 1 + c 2 + y p The general solution for a nonhomogeneous linear n-th order differential equation complementary function particular solution Homogeneous Second Order DEs with Constant Coefficients d x + c y = 0 a y ' ' + b y ' + c y = 0 y = c 1 + c 2 The general solution for a homogeneous linear n-th order differential equation Linear Equations (1A) 26
Complementary Function DEQ d x + c y = g(x) y p particular solution y p + y c general solution nonhomogeneous eq Associated DEQ d x + c y = 0 complementary function c 1 + c 2 general solution homogeneous eq Linear Equations (1A) 27
y c and y p DEQ d x + c y = g(x) y p particular solution y c + y p general solution nonhomogeneous eq c d x 2 + b d y c d x + c y c 0 many such complementary functions c i many possible coefficients p d x 2 + b d y p d x + c y p g(x) only one particular functions coefficients can be determined Linear Equations (1A) 28
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