Notes 16 Linearly Independence and Wronskians

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1 ECE 6382 Fall 2017 David R. Jackson otes 16 Linearly Independence and Wronskians otes are from D. R. Wilton, Dept. of ECE 1

2 Linear Independence Definition: { φ } A set of functions ( x), n = 1, 2,, is linearly independent if the c n = 0, n. n only solution to the set of homogeneous equations is n= 1 c φ ( x) 0 n n A set of functions that is not linearl y independent is said to be linearly dependent. 2

3 Linear Independence (cont.) Assume linear dependence: n= 1 c φ ( x) 0 n n cn 0 Assume cp 0 Then c c c c c φ = φ φ φ φ φ p 1 p+ 1 p p 1 p 1 c + p c p c p c p c p or φ = aφ + a φ + a φ + a φ + a φ p 1 2 p 1 p 1 p+ 1 p+ 1 Conclusion: One of the functions can be written as a combination of the others. 3

4 Linear Independence (cont.) Examples : (sin x,cos x) are linearly independent 2 3 (,, ) x x x 2 2 (,, 2 ) x x x + x are linearly independent are linearly dependent 4

5 Linear Independence (cont.) ote: The classification of linear independence depends in general on the interval for x that is being considered. Example : f ( x) 1, x 0 cos x, x 0 ( ) g x 2, x 0 x + 2, x 0 These two functions are linearly independent on any interval (a, b) that contains positive x values. These two functions are linearly dependent on any interval that contains only negative x values. 5

6 Wronskians and Linear Independence The Wronskian allows us to test for linear independence on some interval. φ1 φ2 φ φ1 φ2 φ φ 1 φ 2 φ φ 1 φ 2 φ W( x) det = φ φ φ φ φ φ ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) We look at whether or not the Wronskian vanishes identically over an interval. Josef Hoëné-Wronski 6

7 Wronskians and Linear Independence (cont.) Assume linear dependence: For some value of p φ = aφ + a φ + a φ + a φ + a φ : p 1 2 p 1 p 1 p+ 1 p+ 1 Then, taking derivatives, we h ave : φ = aφ + a φ + a φ + a φ + a φ p 1 2 p 1 p 1 p+ 1 p+ 1 φ = aφ + a φ + a φ + a φ + a φ p 1 2 p 1 p 1 p+ 1 p+ 1 φ = aφ + a φ + a φ + a φ + a φ p 1 2 p 1 p 1 p+ 1 p+ 1 ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) φ = aφ + a φ + a φ + a φ + a φ p 1 2 p 1 p 1 p+ 1 p+ 1 The pth column of the Wronskian matrix is thus a combination of the other columns, so the determiant is zero. φ φ φ φ φ φ = = ( ) 0 φ φ φ ( 1) ( 1) ( 1) Hence Linear dependence ( ) 0 7

8 Wronskians and Linear Independence (cont.) Interestingly, ( ) 0 Linear dependence Example (Peano): ( ) 2 ( x, xx) 0, but we have linear independence on any interval ( ab, ) that includes the origin! However, if the functions are analytic, then ( ) 0 linear dependence (proof by Peano). Giuseppe Peano 8

9 Wronskians and Linear Independence (cont.) Summary For arbitrary functions: Linear dependence ( ) 0 For analytic functions: Linear dependence ( ) 0 9

10 Wronskians and Linear Independence (cont.) Example Show that φ ( x) = e, φ ( x) = e, φ ( x) = sinkx are linearly dependent. ikx ikx 3 φ φ φ ikx ikx e e sin kx ikx ikx φ φ φ = ike ike k coskx = 0 φ φ φ 2 ikx 2 ikx 2 ke ke k kx 2 ( k ) (Last row is first row! ) sin ote: The three functions are analytic, so W 0 is equivalent to linear dependence. We can also show that any pair of solutions are linearly independent : ikx ikx φ ( x) = e, φ ( x) = e, φ ( x) = sin kx, φ ( x) = coskx

11 Wronskians for SOLDE*s Assume y ( x), y ( x) are two independent solutions of Hence Also, we have Hence y = y + p( x) y + q( x) y = 0 W( x) = = yy yy 2 1 y 1 y 2 W = yy y y + yy yy 2 1 yy 2 1 ( ) y2( py 1 qy1 ) = y py qy 2 ( Wx) ( ) = p y y y y = pw W = W W = W d dx d dx 2 1 px ( ) ( ln W( x) ) ln ( ) = px ( ) *SOLDE: Second-Order Linear Differential Equation 11

12 Wronskians for SOLDEs (cont.) From the last slide : d dx ( Wx) ln ( ) = px ( ) Integrate from x= ato x= x: x W( x) ln W ( x) ln W ( a) = ln = p( x) dx W( a) a p( x) dx W( x) = W( a) e a > 0 x The Wronskian either vanishes for all x( W( a) = 0 ) or for no x ( W( a) 0)! If the two solutions to a SOLDE are linearly independent in one region, they must be linearly independent everywhere. 12

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