# ch2 - Solutions to Homework 6 Math 110 Fall 2006 Prob...

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Solutions to Homework 6. Math 110, Fall 2006. Prob 2.7.1. (a) True. (b) True. (c) False: the roots of the auxiliary polynomial give frequencies of solutions. (d) False: linear combinations of these are solutions too. (e) True: directly from linearity and homogeneity. (f) False unless all roots have multiplicity 1. (g) True: just form the equation p ( D ) y = 0 if p is a given polynomial. Prob 2.7.2. (a) False, only D -invariant subspaces can occur as solution spaces. (b) False, since the space in question is not D -invariant. (c) True: just differentiate both sides of the equation. (d) True: since p ( D ) x = 0, we also get p ( D ) q ( D ) x = q ( D ) p ( D ) x = 0, and likewise p ( D ) q ( D ) y = 0, hence p ( D ) q ( D )( x + y ) = 0. (e) False: Consider p ( t ) = t - 1, and x ( t ) = y ( t ) = e t . Then z ( t ) := x ( t ) y ( t ) = e 2 t , which is not a solution to ( D - I ) 2 z = 0. Prob 2.7.4. By solving the corresponding auxiliary equations, we get: (a) { exp((1 + 5) t/ 2) , exp((1 - 5) t/ 2) } ; (b) { exp( t ) , t exp t, t 2 exp t } ; (c) { 1 , exp( - 2 t ) , exp( - 4 t ) } . Prob 2.7.9. Notive that the product U 1 · · · U n can be written in any order, since every two operators commute. So, if U i x

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