Discrete Mathematics with Graph Theory (3rd Edition) 155

Discrete Mathematics with Graph Theory (3rd Edition) 155 -...

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Section 5.4 153 Adding, we obtain f(x) + 5xf(x) f(x)(1 + 5x) ao + (al + 5ao)x + (a2 + 5al)X2 + ... + (an + 5an_dxn + ... 2 + 3x + 3x 2 + ... + 3xn + ... Therefore, f(x) = (1- 5x + 25x 2 + ... + (-5txn + ... )(2 + 3x + 3x 2 + ... + 3xn + ... ) = 2 - 7x + 38x 2 + ... + [1(3) + (-5)(3) + . .. + (_5)n-l(3) + (_5)n(2)]xn + ... We simplify the term in square brackets. 3[1 + (-5) + (_5)2 + ... + (_5)n-l] + 2(-5t = 3 (_5)n - 1 + 2( -5t, summing a geometric sequence, -5-1 = 2( -5)n + 1 - (-5)n = 1 + 3( -5)n 2 2 and so an = !(1+3( -5)n). According to Section 5.3, the solution has the formpn +qn, where qn is the general solution to the homogeneous relation an = -5an -l and Pn is a particular solution to an = -5an -l + 3. As in (a), we obtain qn = cl(-5)n. For Pn, we try a linear function Pn = a + bn. Substituting in an = -5an -l + 3, we get a + bn = (-5a + 5b + 3) - 5bn, hence, a
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Unformatted text preview: = -5a + 5b + 3, b = -5b. These equations give b = 0, a = !. Thus, Pn + qn = Cl (-5)n +!. Using ao = 2, we get Cl = ~ and so an = ~(-5)n + ! in agreement with our previous solution. 10. (a) f(x) ao + alX + a2x2 + + anx n + 4xf(x) = 4aox + 4alX2 + + 4an_lX n + 3x 2 f(x) = 3aox2 + + 3an_2Xn + Therefore, f(x) -4xf(x) + 3x 2 f(x) = ao + (al -4ao)x + (a2 -4al + 3ao)x2 + ... + (an -4an-l + 3an_2)X n + ... = 2 -3x. 2 -3x So, f(x)(1 -4x + 3x 2 ) = 2 -3x and f(x) = ( )( ). Now write I-x 1-3x 2 -3x A B (A + B) + (-3A -B)x (l-x)(1-3x) = I-x + 1-3x = (l-x)(1-3x) Then A + B = 2, -3A -B = -3; A = !, B = ~ and so = !(1 + x + x 2 + ... + xn + ... ) + ~(1 + 3x + 9x 2 + ... + 3 n xn + ... ) 2 + 5x + ... + (! + ~(3n))xn + ... and so an = ! + ~(3n). (b) f(x) ao + alX + a2x2 + lOxf(x) lOaox + lOalX2 + 25x 2 f(x) 25aox2 + + anx n + + lOan_lX n + + 25an_2Xn +...
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