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Unformatted text preview: Partial Differential Equations Math 442 C13/C14 Fall 2009 Homework 5 Solutions 1. (Strauss 5.2.2.) Show that cos( x ) + cos( x ) is periodic if is a rational number and compute its period. What happens if is not rational? Solution: Let us first notice that if f, g are both periodic with period p , then so is their sum: ( f + g )( x + p ) = f ( x + p ) + g ( x + p ) = f ( x ) + g ( x ) = ( f + g )( x ) . So we simply have to show that cos( x ) , cos( x ) share a period. Note that the periods of cos( x ) are 2 n with n Z , and the periods of cos( x ) are 2 m/ , where m Z . The question is then, do these two sets of numbers share an element, i.e. is there an n and an m so that 2 n = 2 m ? If = p/q , then we have 2 n = 2 mq p , or n = mq p . This has many solutions, choose, for example, m = p, n = q . If is irrational, then this does not work; in fact, solving the first equation gives = m n which is only possible if is rational. 2. Define f ( x ) = x 3 on the interval [0 , 1]. Compute its Fourier sine series and its Fourier cosine series. Solution: The Fourier sine series coefficients are given by B n = 2 L integraldisplay L f ( x ) sin( nx/L ) dx, and in this case L = 1, so we need to compute B n = 2 integraldisplay 1 x 3 sin( nx ) dx. Integrating by parts several times gives B n = ( 1) n 2(6 n 2 2 ) n 3 3 , and then the F.S.S. is n =1 B n sin( nx ). The Fourier cosine series coefficients are given by A n = 2 integraldisplay 1 x 3 cos( nx ) dx....
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