A1_SOL - 1 Signals and Systems Fall 2014 Solution to...

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1 Signals and Systems Fall 2014 Solution to Homework Assignment #1 1. Find the differential equation relating y ( t ) and x ( t ) if Y ( s ) X ( s ) s 2 2 s 3 s 2 4 s 3 . 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 ) ( 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. Given differential equation y 2 y u , It can be rewritten as 2 y y u (*) We firstly solve the equation: 2 0 y y (**) Characteristic equation: 2 0 k gives the root 2 k .
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