ans1 - , 0) + β (0 , , 1 ,-3) . De±ne vectors F 1 = (1 ,...

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Answers to some problems for the 1st midterm problem set 8. Answer: y 1 ( t ) = 1 - (3 - 2 e t - 3 ) H ( t - 3) , y 2 ( t ) = 30(1 - e t - 3 ) H ( t - 3) . (1) 9. Taking the Laplace, get ( s 2 - 3 s + 2) Y ( s ) = 4 s 2 + s - 4 = 4 + s 3 - 4 s 2 s 2 . Y ( s ) = s 3 - 4 s 2 + 4 s 2 ( s - 2)( s - 1) . Partial fractions, s 3 - 4 s 2 + 4 s 2 ( s - 2)( s - 1) = A s - 2 + B s - 1 + C 1 s + C 2 s 2 . Solve for coeFcients: A = - 1 , B = - 1 , C 1 = 3 , C 2 = 2 . Inverse Laplace, y ( t ) = 3 + 2 t - e 2 t - e t . 10. The function f is equal to zero for 0 < t < 2, then f ( t ) = 1 for 2 t 5, then f ( t ) = 0 for 5 t 6, and ±nally, f ( t ) = 4 for t > 6. The Laplace transform is F ( s ) = e - 2 s - e - 5 s + 4 e - 6 s s . 11. (a) We can write ( α, 0 , β + 2 α, - 3 β ) = α (1 , 0 , 2
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Unformatted text preview: , 0) + β (0 , , 1 ,-3) . De±ne vectors F 1 = (1 , , 2 , 0) and F 2 = (0 , , 1 ,-3) . The span the subspace, and they are linearly independent (because the linear combination above can only be made zero by taking α = β = 0. Therefore, F 1 and F 2 are a basis. (b) The dimension is 2. (c) cos θ = ( F 1 · F 2 ) || F 1 |||| F 2 || = 2 √ 5 √ 10 . 1...
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This note was uploaded on 01/12/2010 for the course MATH 421 taught by Professor Nataliakomarova during the Winter '07 term at San Jose State University .

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