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Solutions_F11_ECH157_HW2

Solutions_F11_ECH157_HW2 - Y(s = 1 2(1 2 s 2 2 2 2 s s 2 s...

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y ( t ) = 2 sin ) cos 1 2 + - 3.11 Since convergent and oscillatory behavior does not depend on initial conditions, assume 0 ) 0 ( ) 0 ( ) 0 ( 2 2 = = = x dt dx dt dx a) Laplace transform of the equation gives 3 2 3 ( ) 2 ( ) 2 ( ) ( ) s X s s X s sX s X s s + + = ) 2 3 2 1 )( 2 3 2 1 )( 1 ( 3 ) 1 2 2 ( 3 ) ( 2 3 j s j s s s s s s s s X - + + + + = + + + = Denominator of [ sX ( s )] contains complex factors so that x ( t ) is oscillatory, and denominator vanishes at real values of s = - 1 and - ½ which are all <0 so that x ( t ) is convergent. See Fig. S3.11a. b) 1 2 ) ( ) ( 2 - = - s s X s X s ) 1 ( ) 1 ( 2 ) 1 )( 1 ( 2 ) ( 2 2 + - = - - = s s s s s X The denominator contains no complex factors; x ( t ) is not oscillatory. The denominator vanishes at s =1 0; x ( t ) is divergent. See Fig. S3.11b. c) 1 1 ) ( ) ( 2 3 = + s s X s X s ) 2 3 2 1 )( 2 3 2 1 )( 1 )( )( ( 1 ) 1 )( 1 ( 1 ) ( 3 2 j s j s s j s j s s s s X - - + - + - + = + + = The denominator contains complex factors; x ( t ) is oscillatory. The denominator vanishes at real s = 0, ½; x ( t ) is not convergent. See Fig. S3.11c.
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3-14 d) s s sX s X s 4 ) ( ) ( 2 = + ) 1 ( 4 ) ( 4 ) ( 2 2 + = + = s s s s s s X The denominator of [ sX ( s )] contains no complex factors; x ( t ) is not oscillatory. The denominator of [ sX ( s )] vanishes at s = 0; x ( t ) is not convergent. See Fig. S3.11d. 0 1 2 3 4 5 6 7 8 9 10 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 time x(t) Figure S3.11a. Simulation of X(s) for case a) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 100 200 300 400 500 600 700 time x(t) Figure S3.11b. Simulation of X(s) for case b)
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0 1 2 3 4 5 6 7 8 9 10 -40 -20 0 20 40 60 80 time x(t) Figure S3.11c. Simulation of X(s) for case c) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 2 4 6 8 10 12 14 16 18 time x(t) Figure S3.11d. Simulation of X(s) for case d)
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3-17 3.13 a) 3 2 3 2 (0) (0) 4 with (0) 0 t dx d x dx x e x dt dt dt = = = = Laplace transform of the equation, 1 1 4 - = + s X(s) X(s) s 3 ) 37 .
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