soln_hw4 - Solutions to Homework 4 HW4 pdf Problems 1. i....

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Solutions to Homework 4 HW4 pdf Problems 1. i. e = cos( π ) + i sin( π ) = - 1 (1) ii. e 2 3 i = e 2 (cos(3) - i sin(3)) (2) 2. i. e + e = cos( θ ) + i sin( θ ) + cos( θ ) - i sin( θ ) = 2 cos( θ ) (3) Therefore, cos( θ ) = e + e 2 (4) ii. e - e = cos( θ ) + i sin( θ ) - cos( θ ) + i sin( θ ) = 2 i sin( θ ) (5) Therefore, sin( θ ) = e - e 2 i (6) 3. i. i e it dt = i cos( t ) + i sin( t ) dt = sin( t ) - i cos( t ) = 1 i (cos( t ) + i sin( t )) = 1 i e it (7) ii. i d dt e it dt = e it = i ie it dt (8) 1
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Therefore, i e it dt = 1 i e it (9) Section 3.1 7. The characteristic equation is r 2 - 9 r +9 = 0, it has solutions r = (9 ± 3 5) / 2. Therefore, y = c 1 e (9+3 5) t/ 2 + c 2 e (9 3 5) t/ 2 . (10) 14. The characteristic equation is 2 r 2 + r - 4 = 0, with roots r = - 1 / 4 ± 33 / 4. The general solution is then, y = c 1 e ( 1 33)
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This note was uploaded on 10/23/2009 for the course MATH 22B taught by Professor Hunter during the Winter '08 term at UC Davis.

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soln_hw4 - Solutions to Homework 4 HW4 pdf Problems 1. i....

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