Higher Order ODE's (3A) Young Won Lim 12/27/15

Transcription

1 Higher Order ODE's (3A)

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 The Properties of a Line Equation (1) y = x f (1) = 1 f (2) = f (1+1)=f (1) + f (1) = 2 f (2) = 2 f (3) = f (2+1)=f (2) + f (1) = 3 f (3) = 3 f (4) = f (3+1)=f (3) + f (1) = 4 multiples of a unit y = x f (0.1) = 0.1 f (0.1) = f (0.1 1)=0.1 f (1) = 0.1 f (0.5) = 0.5 f (0.5) = f (0.5 1)=0.5 f (1) = 0.5 f (1.5) = 1.5 f (1.5) = f (1.5 1)=1.5 f (1) = 1.5 fractions of a unit Higher Order ODEs (3A) 3

4 The Properties of a Line Equation (2) multiples of a unit y = x fractions of a unit f (1) = 1 f (2) = 2 f (3) = 3 f (4) = 4 f (0.5) = 0.5 f (1.0) = 1.0 f (1.5) = 1.5 f (2.0) = 2.0 f (0.1) = 0.1 f (0.2) = 0.2 f (0.3) = 0.3 f (0.4) = 0.4 y = x f (x 1 + x 2 ) = f (x 1 ) + f ( x 2 ) f (k x) = k f (x) Higher Order ODEs (3A) 4

5 Linearity Property a( x 1 + x 2 ) a x 2 a x 1 y = a x Additivity f (x 1 + x 2 ) = f (x 1 ) + f ( x 2 ) a (x 1 + x 2 ) = a (x 1 ) + a ( x 2 ) x 1 x 2 x 1 +x 2 y = a x k a x a x Homogeneity f (k x) = k f ( x) x k x a (k x) = k(a x) Higher Order ODEs (3A) 5

6 Linearity & Affinity Linearity Affinity y = a x y = a x + b Translation a x 1 + b a x 2 + b a(x 1 + x 2 ) + b a x 1 + b a k x 1 + b k (a x + b) from linear + -ity. from the Latin, affinis, "connected with" f (x 1 + x 2 ) = f (x 1 ) + f ( x 2 ) f (k x) = k f (x) Additivity Homogeneity Additivity Homogeneity Higher Order ODEs (3A) 6

7 Linear Map Additivity f (x 1 + x 2 ) = f (x 1 ) + f (x 2 ) x 1 f (x) f (x 1 ) x 1 + x 2 f (x) f (x 1 ) + f ( x 2 ) x 2 f (x) f (x 2 ) Homogeneity f (k f (k x) = k f (x) f ( x) x f (x) f (x) k x f (x) k f ( x) Higher Order ODEs (3A) 7

8 Linear Operators Additivity d d f [f + g] = + d g f (x) d f ' (x) f (x) + g( x) d f ' (x ) + g' (x) g( x) d g' (x) Homogeneity d [k f ] = k d f f (k x) = k f ( x) f (x) d f ' (x) k f ( x) d k f '( x) Higher Order ODEs (3A) 8

9 Linear Systems Additivity S{g 1 (x) + g 2 (x)} = S {g 1 ( x)} + S{g 2 ( x)} g 1 (x) S y 1 ( x) g 1 (x) + g 2 ( x) S y 1 ( x) + y 2 ( x) g 2 ( x) S y 2 ( x) d 2 y 2 + a 1 d y + a 2 y = g( x) Homogeneity f (k x) = k f ( x) S{k f (x)} = k S{f (x)} g( x) S y(x) k g( x) S k y(x) Higher Order ODEs (3A) 9

10 Differential Operator Differential Operator D y(x) y '( x) D = d dx D y = d y dx D( y) = d y dx D( y(x)) = d y dx N-th Order Differential Operator y(x) L L( y(x)) L = ( x) D n + 1 ( x) D n a 1 ( x) D + a 0 ( x) L( y) = { (x) D n + 1 (x) D n a 1 ( x) D + a 0 (x)}( y) L( y) = ( x) D n ( y) + 1 ( x) D n 1 ( y) + + a 1 ( x)d( y) + a 0 ( x)( y) L( y) = (x) dn y n + 1 (x) dn 1 y n a 1 ( x) d y + a 0( x) y Higher Order ODEs (3A) 10

11 Examples f (x) d f ' (x) d f = f '(x) D (D f ) = d dx ( d f ) = f ' ' (x) f (x) D f ' (x) D f = f '( x) D 2 f = d2 f 2 = f ' ' (x) Differential Operator : Linear D (c f (x)) = c D f (x) D (f (x) + g(x)) = D f ( x) + D g(x) D (α f ( x) + β g(x)) = α D f ( x) + β D g(x) n-th order Differential Operator L = (x)d n + 1 (x) D n a 1 ( x)d + a 0 (x) (D D + 1) f (x) = D 2 f ( x) + 2 D f ( x) + f (x) = f ' '( x) + 2 f '(x) + f ( x) n-th order Differential Equations using the Differential Operator y' ' + 5 y '+2 y = 3 x D D + 2 = L L( y) = 0 L( y) = g(x) y' ' + 5 y '+2 y = 0 y' ' + 5 y '+2 y = 3 x Higher Order ODEs (3A) 11

12 Linear System Linear System g( x) S y (x) (x) dn y n + 1( x) dn 1 y n a 1( x) d y + a 0( x) y = g(x) ( (x)d n + 1 (x)d n a 1 ( x) D + a 0 (x))( y( x)) = g( x) L( y(x)) = g( x) Higher Order ODEs (3A) 12

13 Linear Differential Equations (x) dn y n + 1( x) dn 1 y n a 1( x) d y + a 0( x) y = g(x) (x) dn y n + 1( x) dn 1 y n a 1( x) d y + a 0( x) y = 0 Non-homogeneous Equation Homogeneous Equation L = ( x) D n + 1 ( x) D n a 1 ( x) D + a 0 ( x) y(x) L L( y(x)) L( y) = ( x) dn y n + 1(x) dn 1 y n a 1(x) d y + a 0( x) y Higher Order ODEs (3A) 13

14 Linear Differential Equations Linear Equation - Additivity (x) dn y 1 n (x) dn y 2 n + 1(x) d n 1 y 1 n a 1(x) d y 1 + a 0(x) y 1 = L( y 1 ) y 1 (x) L(y 1 (x)) + 1(x) d n 1 y a 1(x) d y 2 n 1 + a 0(x) y 2 = L( y 2 ) y 2 (x) L(y 2 (x)) ( x) dn ( y 1 + y 2 ) n + 1 ( x) d n 1 ( y 1 + y 2 ) + + a n 1 1 ( x) d( y 1+ y 2 ) + a 0 (x)( y 1 + y 2 ) = L( y 1 + y 2 ) y 1 + y 2 L(y 1 )+L( y 2 ) Linear Equation - Homogeneity ( x) dn y 1 n + 1(x) d n 1 y a 1(x) d y 1 n 1 + a 0(x) y 1 = L( y 1 ) y 1 (x) L( y 1 (x)) (x) dn k y 1 n + 1 (x) d n 1 k y a n 1 1 (x) d k y 1 + a 0 (x)k y 1 = L(k y) k y 1 ( x) k L( y 1 (x)) Higher Order ODEs (3A) 14

15 Linear Differential Equation Solutions Linear Differential Equation Solution Additivity ( x) dn y 1 n + 1(x) d n 1 y a 1(x) d y 1 n 1 + a 0(x) y 1 = g 1 (x) g 1 (x) y 1 (x) ( x) dn y 2 n + 1(x) d n 1 y a 1(x) d y 2 n 1 + a 0(x) y 2 = g 2 (x) g 2 (x) y 2 (x) ( x) dn ( y 1 + y 2 ) n + 1 ( x) d n 1 ( y 1 + y 2 ) + + a n 1 1 ( x) d( y 1+ y 2 ) + a 0 (x)( y 1 + y 2 ) = g 1 (x) + g 2 (x) Superposition g 1 +g 2 y 1 + y 2 Linear Differential Equation Solution Homogeneity (x) dn y 1 n + 1(x) d n 1 y a 1(x) d y 1 n 1 + a 0(x) y 1 = g 1 (x) g 1 (x) y 1 (x) (x) dn k y 1 n + 1 ( x) d n 1 k y a n 1 1 ( x) d k y 1 + a 0 ( x) k y 1 = k g 1 ( x) k g 1 (x) k y 2 ( x) Higher Order ODEs (3A) 15

16 Homogeneous Equation (x) dn y n + 1( x) dn 1 y n a 1( x) d y + a 0( x) y = g(x) (x) dn y n + 1( x) dn 1 y n a 1( x) d y + a 0( x) y = 0 Associated Homogeneous Equation d n y + a d n 1 y n n a d y n a 0 y = g( x) d n y + a d n 1 y n n a d y n a 0 y = 0 Associated Homogeneous Equation with constant coefficients m n + 1 m n a 1 m + a 0 = 0 Auxiliary Equation m = m 1, m 2,, m n y = c 1 e m 1 x + c 2 e m 2x + + c n e m n x n solutions of the Auxiliary Equation General Solutions of the Homogeneous Equation Higher Order ODEs (3A) 16

17 Non-homogeneous Equation (x) dn y n + 1( x) dn 1 y n a 1( x) d y + a 0( x) y = g(x) (x) dn y n + 1( x) dn 1 y n a 1( x) d y + a 0( x) y = 0 Associated Homogeneous Equation d n y + a d n 1 y n n a d y n a 0 y = g( x) d n y + a d n 1 y n n a d y n a 0 y = 0 Associated Homogeneous Equation with constant coefficients Particular Solution y = c 1 e m 1 x + c 2 e m 2 x + + c n e m n x + y p ( x) General Solutions of the Non-homogeneous Equation Higher Order ODEs (3A) 17

18 Particular Solutions : Variation of Parameters d n y + a d n 1 y n n a d y n a 0 y = g( x) y h (x) = c 1 e m 1 x + c 2 e m 2 x + + c n e m n x k-th column (n 1) y 1 y 1 y 2 y n (1) (1) (1) y 1 y 2 y n (n 1) y 2 y (n 1) y 1 0 y n (1) (1) y 1 0 y n = W = W k uk' (x) = (n 1) n y 1 g(x) y n (n 1) W k W y p ( x) = u 1 e m 1 x + u 2 e m 2 x + + u n e m n x Higher Order ODEs (3A) 18

19 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]