HWSol_Ch9_HW8 - Homework#8 Note 9.33 Given Laplace...

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Homework # 8 Note 9.33 Given Laplace transform H ( s ) and X ( s ) , we have Y ( s )= H ( s ) X ( s )= 2 ( s 2 +2 s +2)( s 1) We need to have the partial fractional expansion for Y ( s ) : 1 Y ( s )= A s 1 + Bs + C s 2 +2 s +2 Assume the partial fractional expression is obtained in Y ( s ) Y ( s )= A s 1 + Bs + C s 2 +2 s +2 = A ( s 2 +2 s +2)+( Bs + C )( s 1) ( s 2 +2 s +2)( s 1) Combine the common items in the numerator, Y ( s )= A ( s 2 +2 s +2)+( Bs + C )( s 1) ( s 2 +2 s +2)( s 1) = ( A + B ) s 2 +(2 A B + C ) s +(2 A C ) ( s 2 +2 s +2)( s 1) Y ( s )= ( A + B ) s 2 +(2 A B + C ) s +(2 A C ) ( s 2 +2 s +2)( s 1) = 2 ( s 2 +2 s +2)( s 1) = 0 s 2 +0 s 2 ( s 2 +2 s +2)( s 1) Accordingly, we have A + B =0 from coefficient of s 2 2 A B + C =0 from coefficient of s 2 A C = 2 from constant item 1 Note that the numerator the each individual expression should have one order lower than the denomina- tor. Otherwise, you will have constant item. For example, if you have As + B s 1 , then it reduces to be A + B + A s 1 But in Y ( s ) , there is no constant item.
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