527solns10

527solns10 - 642:527 SOLUTIONS: ASSIGNMENT 10 FALL 2007...

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Unformatted text preview: 642:527 SOLUTIONS: ASSIGNMENT 10 FALL 2007 Section 17.3: 4 (g) f ( x ) = | sin x | has period (since sin( x + ) =- sin x ) so its Fourier series will have the form FS f = a + summationdisplay n =1 [ a n cos 2 nx + b n sin 2 nx ] . Since f ( x ) is even ( | sin(- x ) | = | - sin x | = | sin x | ) the coefficients b n will all vanish. Moreover, using sin = ( e i- e i ) / 2 i and cos = ( e i + e i ) / 2, we have a = 1 integraldisplay f ( x ) dx = 1 integraldisplay sin( x ) dx = 2 , a n = 2 integraldisplay f ( x )cos(2 nx ) dx = 2 integraldisplay sin( x )cos(2 nx ) dx = 1 2 i integraldisplay bracketleftBig e (2 n +1) ix- e (2 n 1) ix + e ( 2 n +1) ix- e ( 2 n 1) ix bracketrightBig dx =- 1 2 bracketleftbigg e (2 n +1) ix 2 n + 1- e (2 n 1) ix 2 n- 1- e (2 n 1) ix 2 n- 1 + e (2 n +1) ix 2 n + 1 bracketrightbigg =- 4 (4 n 2- 1) . That is, FS f = 2 - 4 summationdisplay n =1 cos 2 nx 4 n 2- 1 . 16 (b) The complex Fourier series of f ( x ) is summationdisplay n = c n e inx with c n = 1 2 integraldisplay 2 f ( x ) e inx dx = 1 2 integraldisplay 2 e (1 in ) x dx = e 2- 1 2(1- in ) . 18 (c) The given function F ( t ) is periodic with period 2; it is even, so there are no sine terms in its Fourier series. The cosine coefficients are a = 1 2 integraldisplay 2 F ( t ) dt = 5 2 , a n = integraldisplay 2 f ( t )cos ntdt = 2 integraldisplay 1 5 t cos nt dt = braceleftbigg- 20 / ( n ) 2 , n odd, , n even so that, since f ( t ) is continuous, f ( t ) = 5 2- summationdisplay n =1 , 3 , 5 ,... 20 n 2 2 cos nt To find the steady-state solution x + x = F ( t ) we observe that the steady-state solution of x + x = 1 is x ( t ) = 1 and that the steady-state solution of x + x = cos( nt ) is x ( t ) = cos( nt ) / (( n ) 2 + 1), and then superimpose: x ( t ) = 5 2- summationdisplay n =1 , 3 , 5 ,......
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This note was uploaded on 09/29/2009 for the course 642 527 taught by Professor Speer during the Fall '07 term at Rutgers.

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527solns10 - 642:527 SOLUTIONS: ASSIGNMENT 10 FALL 2007...

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