samplefinal - F ( s ) = 7 s 2 + 23 s + 30 ( s-2)( s 2 + 2 s...

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MAP 2302 Final Exam Please write your answers in full detail. 1. (6 points) Let f ( t ) be the periodic function of period 2 such that f ( t ) = ± t, 0 t < 1 2 - t, 1 t < 2 . Graph this function and its windowed version. Find the Laplace transform of f ( t ). 2. (5 points) Find the general solution of the equation y 000 + 2 y 00 - 9 y 0 - 18 y = - 18 x 2 - 18 x + 22 . 3. (6 points) Given that f ( x ) = e x is a solution of the equation xy 00 - ( x + 1) y 0 + y = 0 , x > 0 , find a second linearly independent solution. 4. (5 points) If L{ f ( t ) } ( s ) = F ( s ) and L{ g ( t ) } ( s ) = G ( s ) express L{ e t [ f 0 ( t ) * ( g ( t - 5) u ( t - 5))] } ( s ) in terms of F ( s ) and G ( s ), explaining clearly which properties of the Laplace trans- form you use in each step. (Pay attention to the brackets!) 5. (6 points) Find the inverse Laplace transform of
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Unformatted text preview: F ( s ) = 7 s 2 + 23 s + 30 ( s-2)( s 2 + 2 s + 5) . 6. (6 points) Solve the initial value problem y 00-y = 4 ( t-2) + t 2 ; y (0) = 0 ,y (0) = 2 , where ( t ) is the Dirac delta function. 7. (6 points) Solve the initial value problem y 00 + 3 y + 2 y = e-3 t u ( t-2) , y (0) = 2 , y (0) = 0 . Formulae (1) L{ e at f ( t ) } ( s ) = F ( s-a ) (2) L{ f ( t ) } ( s ) = sF ( s )-f (0) (3) L{ tf ( t ) } ( s ) =-d ds F ( s ) (4) L{ f ( t-a ) u ( t-a ) } ( s ) = e-as F ( s ) (5) L{ ( f * g )( t ) } ( s ) = F ( s ) G ( s )...
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samplefinal - F ( s ) = 7 s 2 + 23 s + 30 ( s-2)( s 2 + 2 s...

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