Determine the Laplace transform of:
[tex]u(t-4)[t - 4]^2[/tex]
Answer (2)
The Laplace transform of the function u ( t − 4 ) ( t − 4 ) 2 is s 3 2 e − 4 s . This is derived using the shifting theorem and the basic Laplace transform of the polynomial function t 2 . Understanding the unit step function and the shifting property is crucial for this calculation.
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To determine the Laplace transform of the function u ( t − 4 ) [ t − 4 ] 2 , we can follow these steps step-by-step:
Understand the components of the function :
u ( t − 4 ) is the unit step function that shifts the function by 4 units. It equals 0 for t < 4 and 1 for t ≥ 4 .
( t − 4 ) 2 is simply a quadratic expression, representing the square of t − 4 .
Apply the shifting property of the Laplace transform :
The Laplace transform of f ( t − a ) u ( t − a ) is given by e − a s F ( s ) , where F ( s ) is the Laplace transform of f ( t ) .
Find the Laplace transform of t 2 :
The Laplace transform of t n is s n + 1 n ! for n = 2 , which gives us s 3 2 .
Apply the shifting property :
For u ( t − 4 ) [ t − 4 ] 2 , set a = 4 and use the previously calculated Laplace transform of t 2 , which is F ( s ) = s 3 2 .
The Laplace transform is given by e − 4 s s 3 2 .
So, the Laplace transform of u ( t − 4 ) [ t − 4 ] 2 is e − 4 s s 3 2 .
