secton64shorter

# secton64shorter - Formulas Lcfw f(t s = F s Lcfw_eat f(t s...

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1 Formulas ) ( ) )}( ( ) ( { s F e s c t f t u L cs c = ) ( ) )}( ( { ) ( ) )}( ( { a s F s t f e L or a s F s t f e L at at + = = ) ( ) ( ))} ( ( { )} ( { ) ( 1 1 c t f e t u t f e L e L a s F e L c t a c at cs cs = = + )). 5 ( 4 sin( ) ( 16 ) 3 ( 4 ) 5 ( 3 5 2 5 1 = + + t e t u s e L t s ) ( ) )}( ( { s F s t f L = Examples: . ) 3 ( ) ( ) 5 ( ! 7 7 ) 3 ( 5 3 8 3 1 = t e t u s e L t s ) ( ) ( ))} ( ( { )} ( { ) ( 1 1 c t f e t u t f e L e L a s F e L c t a c at cs cs = = Example 1: Initial Value Problem (1 of 12) c Find the solution to the initial value problem c Such an initial value problem might model the response of a damped oscillator subject to g ( t ), or current in a circuit for a unit voltage pulse. < < = = = = = + + 20 and 5 0 , 0 20 5 , 1 ) ( ) ( ) ( where 0 ) 0 ( , 0 ) 0 ( ), ( 2 2 20 5 t t t t u t u t g y y t g y y y ) ( ) )}( ( ) ( { s H e s c t h t u L cs c =

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2 Example 1: Laplace Transform (2 of 12) c Assume the conditions of Corollary 6.2.2 are met. Then or c Letting Y ( s ) = L { y }, c Substituting in the initial conditions, we obtain c Thus )} ( { )} ( { } { 2 } { } { 2 20 5 t u L t u L y L y L y L = + + [ ] [ ] s e e y L y y sL y sy y L s s s 20 5 2 } { 2 ) 0 ( } { ) 0 ( 2 ) 0 ( 2 } { 2 = + + ( ) ( ) ( ) s e e y y s s Y s s s s 20 5 2 ) 0 ( 2 ) 0 ( 1 2 ) ( 2 2 = + + + ( ) ( ) s e e s Y s s s s 20 5 2 ) ( 2 2 = + + ( ) ( ) 2 2 ) ( 2 20 5 + + = s s s e e s Y s s 0 ) 0 ( , 0 ) 0 ( ), ( ) ( 2 2 20 5 = = = + + y y
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secton64shorter - Formulas Lcfw f(t s = F s Lcfw_eat f(t s...

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