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Notes-LT2 - Step Functions and Laplace Transforms of...

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Step Functions; and Laplace Transforms of Piecewise Continuous Functions The present objective is to use the Laplace transform to solve differential equations with piecewise continuous forcing functions (that is, forcing functions that contain discontinuities). Before that could be done, we need to learn how to find the Laplace transforms of piecewise continuous functions, and how to find their inverse transforms. Step Functions Definition : The unit step function (or Heaviside function ), is defined by < = c t c t t u c , 1 , 0 ) ( , c ≥ 0 . Often the unit step function u c ( t ) is also denoted as u ( t - c ) , H c ( t ) , or H ( t - c ) . The step could also be negative (going down). The complement function is < = - c t c t t u c , 0 , 1 ) ( 1 , c ≥ 0 .
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The Laplace transform of the unit step function is L { u c ( t )} = s e cs - , s > 0, c ≥ 0 Notice that when c = 0, u 0 ( t ) has the same Laplace transform as the constant function f ( t ) = 1. (Why?) Therefore, for our purpose, u 0 ( t ) = 1. (Keep in mind that the Laplace transforms method only works for t ≥ 0.) Note : The calculation of L { u c ( t )} goes as follow (given that c ≥ 0): L { u c ( t )} = - - - - = = c c st st st c e s dt e dt e t u 1 1 ) ( 0 ( ) s e e s cs cs - - = - - = 0 1 , s > 0.
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The unit step function is much more useful than it first appears to be. When in a product with a second function, the unit step function acts like a switch to turn the other function on or off: < = c t t f c t t f t u c ), ( , 0 ) ( ) ( , (an “on” switch) < = - c t c t t f t f t u c , 0 ), ( ) ( )) ( 1 ( , (an “off” switch). By combining two unit step functions, we can also selectively make a function appears only for a finite duration, then it disappears. That is, the function is switched “on” at a , then is switched “off” at a later time b . < < = - b t b t a t f a t t f t u t u b a , 0 ), ( , 0 ) ( )) ( ) ( ( , where 0 ≤ a < b . We could think this combination as an “on-off” toggle switch that controls the appearance of the second function f ( t ). By cascading the above types of products, we can now write any piecewise- defined function in a succinct form in terms of unit step functions.
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Suppose < < < = d t t f c t b t f b t a t f a t t f t F n ), ( : : ), ( ), ( ), ( ) ( 3 2 1 . Then, we can rewrite F ( t ), succinctly, as F ( t ) = (1 - u a ( t )) f 1 ( t ) + ( u a ( t ) - u b ( t )) f 2 ( t ) + ( u b ( t ) - u c ( t )) f 3 ( t ) + … + u d ( t ) f n ( t ). Example : < + < - = 9 ), 2 cos( 9 4 , 4 , 2 3 ) ( 5 2 t t t t e t t t F t . Then, F ( t ) = (1 - u 4 ( t )) (3 t 2 - 2) + ( u 4 ( t ) - u 9 ( t )) ( e 5 t + t ) + u 9 ( t ) cos(2 t ).
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The difference between u c ( t ) f ( t ) and u c ( t ) f ( t - c ) Example : u π /2 ( t ) sin( t ) and u π /2 ( t ) sin( t - π /2) Fig. Graph of : u π /2 ( t ) sin( t ) Fig. Graph of : u π /2 ( t ) sin( t - π /2)
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The effects of the Laplace transform on translations Arguably the most important formula for this class, it is usually called the Second Translation Theorem (of the Laplace transform): Theorem : If F ( s ) = L { f ( t )}, and if c is any positive constant, then L { u c ( t ) f ( t - c )} = e - cs L { f ( t )} = e - cs F ( s ).
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