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Assignment1-Solutions

# Assignment1-Solutions - ECE3085 Solution#1 Problem 1(5 pts...

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ECE3085 - Solution #1 Problem 1. (5 pts) What is the Laplace transform of f ( t ) = α sin( ωt ) + β cos( t ) + e - αt sin( ωt ) ? Solution 1. Using the linear properties of the Laplace transform we proceed as follows: L [ f ( t )] = α L [sin( ωt )] + β L [cos( t )] + L [ e - αt sin( ωt )] . = αω s 2 + ω 2 + βs 2 s 2 + 1 + ω ( s + α ) 2 + ω 2 Problem 2. (5 pts) What is the Laplace transform of f ( t ) = ( t + 1) 2 ? Solution 2. The best thing to do is to expand out the square and then solve for each component individually using the Linearity property. Since the individual components are polynomials in time, we can apply the Multiplication by Time property. L [ f ( t )] = L [( t + 1) 2 ] = L [ t 2 + 2 t + 1] = L [ t 2 ] + 2 L [ t ] + L [1] = 2 s 3 + 2 s 2 + 1 s = 2 + 2 s + s 2 s 3 = s 2 + 2 s + 2 s 3 Problem 3. (10 pts) Given that f ( t ) and F ( s ) form a Laplace transform pair, what is the Laplace transform of the following related functions: a. g ( t ) = integraltext t 0 integraltext σ 1 0 f ( σ 2 )d σ 2 d σ 1 . b. g ( t ) = f ( t )cos( ωt ) . Solution 3. a. This solution is basically the application of the integral rule twice. L [ g ] = L bracketleftbiggintegraldisplay t 0 integraldisplay σ 1 0 f ( σ 2 )d σ 2 d σ 1 bracketrightbigg = 1 s L bracketleftbiggintegraldisplay t 0 f ( σ )d σ bracketrightbigg = 1 s parenleftbigg 1 s L [ f ( t )]

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