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SampleTest1v2

# SampleTest1v2 - L f = 1 1 1 1 1 1 1 1 1 1 2 1 2 1 2 2 2 s s...

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MATH3705 Solution to Test 1 Summer 2006 1 SOLUTION TO TEST 1 MATH3705, Summer 2006 1. (2 marks) Express the following rational function as a sum of partial fractions: F ( s ) = ) 1 )( 1 ( 2 + - s s s . Solution . F ( s ) = [1 / ( s - 1) - s / ( s 2 + 1) + 1 / ( s 2 + 1)] / 2. 2. (2 marks) Find the inverse Laplace transformation of F ( s ) = 1 / ( s - 1) - s / ( s 2 + 1) + 1 / ( s 2 + 1). Solution . L [ F ( s )] = e t - cos t + sin t . 3. (5 marks) Use the definition of Laplace transformation to find the Laplace transformation of the function f ( t ) = t e t . Solution . L [ f ] = ( 1) ( 1) ( 1) 0 0 0 0 1 1 1 1 t st s t s t s t te e dt t de e dt te s s - - - - - - - = - = - - - ( 1) ( 1) 2 2 0 0 1 1 1 1 ( 1) ( 1) s t s t e dt e s s s - - - - = = - = - - - . 4. (5 marks) Find the Laplace transformation of the function = + < + = + < = ... , 2 , 1 , 0 , 2 2 1 2 0 ... , 2 , 1 , 0 , 1 2 2 1 ) ( k k t k k k t k t f The graph of this function is as follows: Solution . This function has a fundamental period 2.

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Unformatted text preview: L [ f ] = [ ] ) 1 ( 1 ) 1 ( 1 ) 1 ( 1 1 1 ) ( 1 1 2 1 2 1 2 2 2 s s s st s st s st s e s e s e e e s dt e e dt e t f e---------+ =--=--=-=-âˆ« âˆ« . 5. (6 marks) Use Laplace transformation to solve the differential equation t f ( t ) MATH3705 Solution to Test 1 Summer 2006 2 y" + y = e t , y (0) = 0, y' (0) = 1. Solution . Let Y ( s ) = L [ y ]. Take the Laplace transformation on both sides: s 2 Y-1 + Y = 1 / ( s-1). ( s 2 + 1) Y = s / ( s-1). Y = s / [( s-1)( s 2 + 1)] = (1 / 2)(1 / ( s-1) -s / ( s 2 + 1) + 1 / ( s 2 + 1)). Take the inverse Laplace transformation, using the result in Question 1 and 2, y = (1 / 2)( e t-cos t + sin t )....
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