A1_SOL - Signals and Systems Fall 2013 Solution to Homework...

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Signals and Systems Fall 2013 Solution to Homework Assignment #1 1. Find the differential equation relating y ( t ) and x ( t ): 3 2 2 2 ( ) ( ) 4 ( ) 3 ( ) ( ) 2 ( ) s Y s s Y s sY s Y s s X s s X s + + = + Taking inverse Laplace transform of the above equation, leads to: ( ) ( ) 4 ( ) 3 ( ) ( ) 2 ( ) y t y t y t y t x t x t + + = +    . 2. With the given equation: = + t d x t x t y t y 0 ) ( ) ( ) ( 3 ) ( τ τ Taking Laplace transform: 0 ( ) ( ) (0) 3 ( ) ( ) X s sY s y Y s X s s = + = ( ) ( ) 1 3 ( ) ( ) s s Y s X s s + = ( ) 2 1 ( ) ( ) ( ) 3 s Y s H s X s s s = = + . 3. Solve the differential equation ) ( ) ( 9 ) ( t f t y t y + = with initial conditions y (0) = 0, ) 0 ( y = 6, and with f ( t ) = 0. ) ( ) ( 9 ) ( t f t y t y + = ( ) 9 ( ) 0 y t y t + =  (*) Solving ( ) 9 ( ) 0 y t y t + =  (**) Characteristic equation: 2 9 0 k + = , gives the complex roots are 3 k j = ± . Hence, the solution of (**) has the form of: 1 2 cos3 sin3 H y C t C t = +
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