Second Order ODE's (2A) Young Won Lim 5/5/15

Transcription

1 Second Order ODE's (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 Homogeneous Linear Equations with constant coefficients Second Order ODEs (2A) 3

4 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) Second Order ODEs (2A) 4

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) Second Order ODEs (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 Second Order ODEs (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 Second Order ODEs (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 Second Order ODEs (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 Second Order ODEs (2A) 9

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 Second Order ODEs (2A) 10

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 Second Order ODEs (2A) 11

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 Second Order ODEs (2A) 12

13 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) Second Order ODEs (2A) 13

14 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)] Second Order ODEs (2A) 14

15 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 Second Order ODEs (2A) 15

16 Reduction of Orders Second Order ODEs (2A) 16

17 Finding another solution from the known Second Order EQ a d2 y d x + b d y d x + c y = 0 1 = f (x) known solution = u(x)f (x) another solution to be found We know one solution Suppose the other solution (x) = e m 1 x = e m 2 x = e b 2 a x (x) = u(x) (x) = u(x)e m 1 x Condition for (t) to be a solution Find u(x) Second Order ODEs (2A) 17

18 Conditions for to be another solution a d2 y d x 2 + b d y d x + c y = 0 a y ' ' + b y ' + c y = 0 = u ' = u' + u ' ' ' = u' ' + 2u' ' + u ' ' a ' ' + b ' + c = 0 a[u' ' + 2 u' ' + u ' '] + b[u' + u '] + c u = 0 a ' ' + b ' + c = 0 u[a ' ' + b ' + c ] + a[u ' ' + 2u ' '] + b[u' ] = 0 Condition for (t) to be a solution (x) = u(x) (x) a u' ' + u' [2a ' + b ] = 0 Second Order ODEs (2A) 18

19 Reduction of Order We know one solution (x) Suppose the other solution (x) = u(x) (x) a d2 y d x 2 + b d y d x + c y = 0 a y ' ' + b y ' + c y = 0 a y ' ' 2 + b y ' 2 + c = 0 au' ' + u' [2 a ' + b ] = 0 2 nd Order w(x) = u '(x) a w' + w [2 a ' + b ] = 0 1 st Order u = c 1 e (b /a)x 2 dx + c 2 = c 1 e (b/a) x 2 dx + c 2 (c 1 =1, c 2 =0) = e (b /a) x 2 dx Second Order ODEs (2A) 19

20 General Solutions for the repeated roots case = e (b /a) x 2 dx m 1 = ( b + b 2 4ac)/2 a m 2 = ( b b 2 4ac)/2 a b 2 4ac = 0 m 1 = b/2a m 2 = b/2a e m 1 x = e m 2x = e b 2 a x (x) = e b 2 a x 2 = e b a x = e b 2a x a)x b e (b/ 2a dx = x e (b/ a)x 1dx e (x) = e b 2 a x (x) = x e b 2a x ' ' y 0 y (x) = c 1 (x) + c 2 (x) 2 Second Order ODEs (2A) 20

21 General Solutions - Homogeneous Equation - Non-homogeneous Equation Second Order ODEs (2A) 21

22 General Solution Homogeneous Equations 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 (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, m 2 y = C 1 e m 1x + C 2 e m 2 x (B) b 2 4ac = 0 Real, equal m 1, m 2 y = C 1 e m 1x + C 2 x e m 1x (C) 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 = e α x (C 3 cos(β x) + C 4 sin (β x)) Second Order ODEs (2A) 22

23 Complementary Function DEQ a d2 y d x 2 + b d y d x + c y = g(x) particular solution + y c a complementary function that makes a general (whole) solution the general solution of a nonhomogeneous eq Associated DEQ a d2 y d x 2 + b d y d x + c y = 0 c 1 + c 2 homogeneous solution the general solution of a homogeneous eq Second Order ODEs (2A) 23

24 y c and DEQ a d2 y d x 2 + b d y d x + c y = g(x) particular solution y c + general solution nonhomogeneous eq a d2 y c d x 2 + b d y c d x + c y c 0 y c = c 1 e m 1 x + c 2 e m 2 x many such complementary functions c i manossible coefficients a d2 d x 2 + b d d x + c g(x) = a similar form only one particular functions coefficients can be determined Second Order ODEs (2A) 24

25 Finding a Particular Solution - Undetermined Coefficients Second Order ODEs (2A) 25

26 Particular Solutions DEQ a d2 y d x 2 + b d y d x + c y = g(x) When coefficients are constant 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 Second Order ODEs (2A) 26

27 Form Rule DEQ a d2 y d x 2 + b d y d x + c y = g(x) When coefficients are constant particular solution by a conjecture (I) FORM Rule (II) Multiplication Rule g(x) = 2 = A g(x) = 3 x+4 = Ax+B g(x) = 6 x 2 7 = Ax 2 +Bx+C g(x) = sin 8 x = A cos 8 x+ B sin 8 x g(x) = cos 9 x = A cos 9 x+ B sin 9 x g(x) = e 10 x = A e 10 x g(x) = x e 11 x = ( Ax+ B)e 11 x g(x) = e 11 x sin 12 x = A e 11 x sin 12x+ B e 11 x cos12 x g(x) = 5 x sin (3 x) = ( Ax+B)cos(3 x) + (Cx+D)sin (3 x) Second Order ODEs (2A) 27

28 Form Rule Example DEQ a d2 y d x 2 + b d y d x + c y = g(x) + y c = Ax+B '= A ' '=0 ' ' + 3 ' + 2 = 3 A + 2(A x+b) = 2 A x+3 A+2 B = x Associated DEQ a d2 y d x 2 + b d y d x + c 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 = 1 2 x 3 4 c 1 + c 2 y = c 1 e x + c 2 e 2x x 3 4 Second Order ODEs (2A) 28

29 Multiplication Rule DEQ a d2 y d x 2 + b d y d x + c y = g(x) use =x n =x n if = = + y c When contains a term which is the same term in y c Use multiplied by x n Associated DEQ a d2 y d x 2 + b d y d x + c y = 0 n is the smallest positive integer that eliminates the duplication c 1 + c 2 Second Order ODEs (2A) 29

30 Multiplication Rule Example (1) y ' ' 2 y '+ y = 2 e x y ' ' 2 y '+ y = 6 x e x = A e x A x e x A x 2 e x = A x e x A x 2 e x A x 3 e x 2 A e x 6 x e x = e x = x e x = e x = x e x y ' ' 2 y '+ y = 0 y ' ' 2 y '+ y = 0 = x (A x+b)e x B x e x = x 2 ( A x+b)e x Second Order ODEs (2A) 30

31 Multiplication Rule Example (2) y '+4 y = e x sin(2 t) + 2 t cos(2 t) (t ) = e x ( A cos(2 t)+b sin(2 t)) + (C t +D)cos(2 t) + (E t +F)sin(2 t) (t ) = e x ( A cos(2 t)+b sin(2 t)) + t (C t +D)cos(2t ) + t (E t +F)sin(2t) y h (t) = c 1 e +i 2 t i 2t +c 2 e = (c 3 cos(2t)+c 4 sin(2t)) y ' '+5 y '+6 y = t 2 e 3 t (t ) = ( A t 2 +B t +C )e 3t (t ) = t ( A t 2 +B t+c)e 3 t y h = c 1 e 2t +c 2 e 3 t Second Order ODEs (2A) 31

32 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 d 2 y c d x 2 + b d y c d x + c y c = 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 1 2 d 2 1 d x 2 + b d 1 d x + c 1 = (2 x2 + 3) d 2 y P 2 d x 2 + b d 2 d x + c y P 2 = cos8 x d 2 d x 2 [ y c ] + b d d x [ y c ] + c[ y c ] = 2 x cos8 x Second Order ODEs (2A) 32

33 Superposition (2) DEQ a d2 y d x 2 + b d y d x + c y = (2 x2 + 3) cos 8 x = ( A x 2 +B x+c) (cos8 x+sin 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 1 2 d 2 d x 2 [ y c+ 1 2 ] + b d d x [ y c+ 1 2 ] + c [ y c ] = (2 x 2 + 3) cos8 x Second Order ODEs (2A) 33

34 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} Second Order ODEs (2A) 34

35 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 Second Order ODEs (2A) 35

36 Finding a Particular Solution - Variation of Parameters Second Order ODEs (2A) 36

37 Variation of Parameter [c u(x)] y ' + P(x) y = 0 y = c e P(x)dx y ' + P(x) y = Q(x) Integrating factor 1 = e + P( x)dx y h = c = u(x) = e P( x)dx y ' ' + P(x) y ' + Q(x) y = 0 y ' ' + P(x) y ' + Q(x) y = f (x) y h = c 1 = u 1 (x) + c 2 + u 2 (x) Second Order ODEs (2A) 37

38 Variation of Parameter [c u(x)] y ' ' + P(x) y ' + Q(x) y = 0 y ' ' + P(x) y ' + Q(x) y = f (x) y h = c 1 = u 1 (x) + c 2 + u 2 (x) = u 1 ' = u 1 ' + u 1 ' + u 2 + u 2 ' + u 2 ' ' ' + P(x) ' + Q(x) = ' ' = u 1 ' ' + u 1 ' ' + u 1 ' ' + u 1 ' ' + u 2 ' ' + u 2 ' ' + u 2 ' ' + u 2 ' ' not a matrix notation + u 1 ' ' + u 1 ' ' + u 1 ' ' + u 1 ' ' + u 2 ' ' + u 2 ' ' + u 2 ' ' + u 2 ' ' + u 1 ' + u 1 ' + P + Q + u 2 ' + u 2 ' + u 1 + u 2 u 1 ( ' ' + P ' + Q ) = 0 u 2 ( ' ' + P ' + Q ) = 0 Second Order ODEs (2A) 38

39 Variation of Parameter [c u(x)] not a matrix notation ' ' + P(x) ' + Q(x) = + u 1 ' ' + u 1 ' ' + u 1 ' ' + u 2 ' ' + u 2 ' ' + u 2 ' ' + P + u 1 ' + u 2 ' = + u 1 ' ' + u 1 ' ' + u 2 ' ' + u 2 ' ' + P + u 1 ' + u 2 ' + + u 1 ' ' + u 2 ' ' = d d x + u 1 ' + u 2 ' + P + u 1 ' + u 2 ' + + u 1 ' ' + u 2 ' ' = f (x) u 1 ' + u 2 ' = 0 u 1 ' ' + u 2 ' ' = f (x) u 1 ' + u 2 ' = 0 ' u 1 ' + ' u 2 ' = f (x) [ ' '][ u 1' '] u = [ 0 2 f (x)] ' y2 ' 0 f (x) u 1 ' = ' y1 0 ' f u 2 (x) ' = ' ' Second Order ODEs (2A) 39

40 Variation of Parameter [c u(x)] y ' ' + P(x) y ' + Q(x) y = 0 y ' ' + P(x) y ' + Q(x) y = f (x) y h = c 1 + c 2 = u 1 (x) + u 2 (x) If the associated homogeneous solution can be solved then, always a particular solution can be found No restriction constant coefficients A constant or A polynomial or An exponential function or A sine and cosine functions or [ ' '][ u 1' '] u = [ 0 2 f (x)] 0 f (x) u 1 ' = ' y 2 ' = ' W 1 W 0 ' f u 2 (x) ' = ' = ' W 2 W ' ' y = W 2 u 1 ' (x) = (x)f (x) W u 2 '(x) = (x)f (x) W Second Order ODEs (2A) 40

41 Homogeneous Linear Equations with variable coefficients Second Order ODEs (2A) 41

42 Cauchy-Euler Equation 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 2 y ' ' + a 1 y ' + a 0 y = g(x) Second Order Linear Equations with Variable Coefficients 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) Cauchy-Euler Equation a 2 x 2 d 2 y d x 2 + a 1 x d y d x + a 0 y = g(x) a 2 x 2 y ' ' + a 1 x y ' + a 0 y = g(x) Second Order ODEs (2A) 42

43 Auxiliary Equation of Cauchy-Euler Equation Homogeneous Second Order Cauchy-Euler Equation a x 2 d 2 y d x 2 + b x d y d x + c y = 0 a x 2 y ' ' + b x y ' + c y = 0 try a solution y = x m a x 2 d 2 d x 2 {xm } + b x d d x {xm } + c {x m }= 0 a {m(m 1)x m } + b{m x m } + c {x m }= 0 (a m 2 + (b a)m + c) x m = 0 a x 2 {x m }' ' + b x {x m }' + c {x m }= 0 a {m(m 1)x m } + b{m x m } + c {x m }= 0 (a m 2 + (b a)m + c) x m = 0 auxiliary equation (a m 2 + (b a)m + c) = 0 (a m 2 + (b a)m + c) = 0 Second Order ODEs (2A) 43

44 General Solution y h of Cauchy-Euler Equations Homogeneous Second Order Cauchy-Euler Equation a x 2 d 2 y d x 2 + b x d y d x + c y = 0 a x 2 y ' ' + b x y ' + c y = 0 try a solution y = x m auxiliary equation (a m 2 + (b a)m + c) = 0 m 1 = { (b a) + (b a) 2 4 ac}/2 a m 2 = { (b a) (b a) 2 4 ac}/2 a (A) (b a) 2 4a c > 0 Real, distinct m 1, m 2 y = C 1 x m 1 + C 2 x m 2 (B) (b a) 2 4a c = 0 Real, equal m 1, m 2 y = C 1 x m 1 + C 2 x m 1 ln x (C) (b a) 2 4 a c < 0 Conjugate complex m 1, m 2 y = C 1 x m 1 + C 2 x m 2 = C 1 x (α+i β) (α i β) + C 2 x Second Order ODEs (2A) 44

45 Complex Exponential Conversion (Cauchy-Euler Equation) Homogeneous Second Order Cauchy-Euler Equation a x 2 d 2 y d x 2 + b x d y d x + c y = 0 a x 2 y ' ' + b x y ' + c y = 0 m 1 = { (b a) + 4 ac (b a) 2 i}/2a m 2 = { (b a) 4 ac (b a) 2 i}/2 a m 1 = α + iβ m 2 = α iβ = x m 1 = x α+i β = x m 2 = x α i β x m 1 = x α x +i β = x α e +i βln x x = e ln x x m 2 = x α x iβ = x α e i β ln x x i β +i β ln x = e y = C 1 x m 1 + C 2 x m 2 = C 1 x α e +i β ln x + C 2 x α +iβ ln x e Pick two homogeneous solution = x α {e +iβ ln x + e iβ ln x }/2 = x α cos(βln x) = x α {e +i β ln x e i β ln x }/2i = x α sin(βln x) (C 1 =+1/2, C 2 =+1/2) (C 1 =+1/2i, C 2 = 1/2 i) y = C 3 x α cos(βln x) + C 4 x α sin(βln x) = x α (C 3 cos(β ln x) + C 4 sin(βln x)) Second Order ODEs (2A) 45

46 Conditions for (Cauchy-Euler Equation) a x 2 d 2 y d x 2 + b x d y d x + c y = 0 a x 2 y ' ' + b x y ' + c y = 0 a x 2 ' ' + b x ' + c = 0 a x 2 ' ' + b x ' + c = 0 = u ' = u' + u ' ' ' = u' ' + 2 u' ' + u ' ' a x 2 [u' ' + 2 u ' ' + u ' '] + b x[u' + u '] + c u = 0 u[a x 2 ' ' + b x ' + c ] + a x 2 [u ' ' + 2u' '] + b x[u' ] = 0 Condition for (t) to be a solution (x) = u(x) (x) a x u ' ' + u'[2 a x ' + b ] = 0 Second Order ODEs (2A) 46

47 Reduction of Order (Cauchy-Euler Equation) We know one solution (x) Suppose the other solution (x) = u(x) (x) a x 2 d 2 y d x 2 + b x d y d x + c y = 0 a x 2 y ' ' + b x y ' + c y = 0 a x 2 ' ' + b x ' + c = 0 a x u' ' + u '[2 a x ' + b ] = 0 w(x) = u '(x) a x w ' + w[2a x ' + b ] = 0 u = c 1 x (b /a) dx + c 2 2 = x (b /a) dx 2 1 = x (b a) a u = c 1 ln x + c 2 = c 1 x (b/a) 2 dx + c 2 (c 1 =1, c 2 =0) = ln x Second Order ODEs (2A) 47

48 Constant v.s. Non-constant Coefficients 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 (A) b 2 4ac > 0 y = C 1 e m 1 x + C 2 e m 2 x (B) b 2 4ac = 0 y = C 1 e m 1 x + C 2 x e m 1 x (C) b 2 4ac < 0 y = C 1 e α x e +iβ x + C 2 e α x e iβ x = e α x (C 3 cos(β x) + C 4 sin(β x)) Homogeneous Second Order Cauchy-Euler Equation a x 2 d 2 y d x 2 + b x d y d x + c y = 0 a x 2 y ' ' + b x y ' + c y = 0 x = e ln x x i β +i β ln x = e (A) (b a) 2 4a c > 0 y = C 1 x m 1 + C 2 x m 2 (B) (b a) 2 4a c = 0 y = C 1 x m 1 + C 2 x m 1 ln x (C) (b a) 2 4 a c < 0 y = C 1 x α e +iβ ln x + C 2 x α +i β ln x e = x α (C 3 cos(β ln x) + C 4 sin(βln x)) Second Order ODEs (2A) 48

49 A Unifying View Constant Coefficients a d2 y d x 2 + b d y d x + c y = 0 Non-constant Coefficients a x2 d 2 y d x 2 + b x d y d x + c y = 0 x = e ln x x i β +i β ln x = e (A) (B) (C) y = C 1 x m 1 + C 2 x m 2 y = C 1 x m 1 + C 2 x m 1 ln x y = C 1 x α e +iβ ln x + C 2 x α +i β ln x e = x α (C 3 cos(β ln x) + C 4 sin(βln x)) (A) (B) y = C 1 e m 1 x + C 2 e m 2 x y = C 1 e m 1 x + C 2 x e m 1 x x (A) (B) y = C 1 e m 1 ln x + C 2 e m 2 ln x y = C 1 e m 1 ln x + C 2 e m 1 ln x ln x ln x (C) y = C 1 e α x e +iβ x + C 2 e α x e iβ x (C) y = C 1 e α ln x e +iβ ln x + C 2 e α ln x +i β ln x e = e α x (C 3 cos(β x) + C 4 sin(β x)) = e α ln x (C 3 cos(β ln x) + C 4 sin(β ln x)) Second Order ODEs (2A) 49

50 Green's Function Second Order ODEs (2A) 50

51 Initial Value Problems y ' ' + P(x) y ' + Q(x) y = 0 y ' (x 0 ) = y ' (x 0 ) = 0 y (x 0 ) = y 0 y (x 0 ) = 0 y ' ' + P(x) y ' + Q(x) y = f (x) ' ' y = W 2 y h = c 1 + c 2 = u 1 (x) + u 2 (x) (x 0 ) = 0 u 1 '(x) = W 1 W = (x)f (x) W u 2 ' (x) = W 2 W = (x)f (x) W u 1 (x) = u 1 ' (x) dx u 2 (x) = u 2 ' (x) dx anti-derivative = (t) f (t ) W (t) dt + c 1 = (t) f (t ) anti-derivative W (t) dt + c 2 x = x 0 (t) f (t) W (t) dt x = x 0 (t) f (t) W (t) dt u 1 (x 0 ) = 0 u 1 (x 0 ) (x 0 ) = 0 u 2 (x 0 ) = 0 u 2 (x 0 ) ( x 0 ) = 0 Second Order ODEs (2A) 51

52 Green's Function and IVP's (1) y ' ' + P(x) y ' + Q(x) y = f (x) [x 0, x] I u 1 ' (x) = (x)f (x) W (x) x y u 1 ( x) = x0 2 (t )f (t) d t W (t ) ' ' y = W (x) 2 u 2 '(x) = (x)f (x) W (x) x u 2 (x) = x 0 (t)f (t) W (t) d t = u 1 (x) + u 2 (x) = [ x 0 = [ x 0 = [ x y x0 x (t)f (t) W (t) x (x) (t) W (t ) d t ] (x) + [ x 0 f (t)d t ] + [ x 0 1 (t ) (x) (x) y (t) 2 ] f (t )d t W (t ) x x (t)f (t) ] d t W (t) y (x) 2 (t) (x) ] f (t)d t W (t ) = x0 x G(x, t)f (t)d t Second Order ODEs (2A) 52

53 Green's Function and IVP's (2) y ' ' + P(x) y ' + Q(x) y = 0 y ' (x 0 ) = y ' (x 0 ) = 0 y (x 0 ) = y 0 y (x 0 ) = 0 y ' ' + P(x) y ' + Q(x) y = f (x) ' ' y = W 2 y h = c 1 + c 2 = u 1 (x) + u 2 (x) x x = u 1 (x) + u 2 (x) = [ (t ) (x) (x) y (t) 2 ] = x0 G(x, t)f (t)d t x0 f (t )d t W (t ) at the end, this x will replace the literal t = x0 x G(x, t)f (t)d t this x and t appear in the indefinite integral Second Order ODEs (2A) 53

54 Green's Function G(x, t ) = [ (t) (x) (x) (t) W (t) ] W (t) = (t ) (t) ' (t) ' (t ) y ' ' + P(x) y ' + Q(x) y = 0, y ' ' + P(x) y ' + Q(x) y = f (x) the same Green's function y ' ' + P(x) y ' + Q(x) y = g(x) G(x, t ) = [ (t) (x) (x) (t) W (t) ] y ' ' + P(x) y ' + Q(x) y = h(x) Second Order ODEs (2A) 54

55 Three Initial Value Problem y ' ' + P(x) y ' + Q(x) y = f (x) y (x 0 ) = y 0 y '(x 0 ) = y ' ' + P(x) y ' + Q(x) y = 0 y (x 0 ) = y 0 y '(x 0 ) = Homogeneous DEQ Nonhomogeneous Initial Conditions Nonzero Initial Conditions y ' ' + P(x) y ' + Q(x) y = f (x) Nonhomogeneous DEQ y (x 0 ) = 0 y '(x 0 ) = 0 Zero Initial Conditions Initially at rest Rest Solution Second Order ODEs (2A) 55

56 General Solutions of the Initial Value Problem y ' ' + P(x) y ' + Q(x) y = f (x) y (x 0 ) = y 0 y '(x 0 ) = y = y h + y (x 0 ) = y h (x 0 ) + (x 0 ) = y = y 0 y '(x 0 ) = y h '(x 0 ) + '(x 0 ) = + 0 = y ' ' + P(x) y ' + Q(x) y = 0 y (x 0 ) = y 0 y '(x 0 ) = y h Nonhomogeneous Initial Conditions Nonzero Initial Conditions Response due to the initial conditions y ' ' + P(x) y ' + Q(x) y = f (x) y (x 0 ) = 0 y '(x 0 ) = 0 x = x0 G(x, t)f (t)d t Zero Initial Conditions Initially at rest Rest Solution Response due to the forcing function f Second Order ODEs (2A) 56

57 Rest Solution y ' ' + P(x) y ' + Q(x) y = f (x) y (x 0 ) = 0 y '(x 0 ) = 0 Nonhomogeneous DEQ Zero Initial Conditions Initially at rest Rest Solution = u 1 (x) + u 2 (x) = [ x y x0 1 (t ) (x) (x) y (t) 2 ] W (t ) f (t )d t x = x0 G(x, t)f (t )d t (x) = x0 x G(x, t)f (t)d t '(x) = G(x, x)f (x) + x0 x x [G(x, t)f (t)] d t = [ x y x0 1 (t ) ' (x) '(x) y (t) 2 ] W (t ) f (t)d t (x 0 ) = x0 x 0 G(x, t )f (t)d t = 0 x 0 '(x 0 ) = [ y x0 1 (t) '(x 0 ) '(x 0 ) (t) W (t) ] f (t)d t = 0 Second Order ODEs (2A) 57

58 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]