041907-132A-clab 3

041907-132A-clab 3 - = 6. (b) Plot cos 2 t from t = 0 to t...

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C HEMICAL E NGINEERING 132A Professor Todd Squires Spring Quarter, 2007 Computer Lab Assignment 3 Laplace Transforms E XERCISES Relevant Commands: LaplaceTransform[y[t],t,s] InverseLaplaceTransform[y[s],s,t] UnitStep[t-10] DiracDelta[t-10] Integrate[x^2,{x,0,3}] Solve[{x^2+2 y ==3,y + 3 x == 0},{x,y}] DSolve[{y’’[x] == ay’[x] + y[x], y[0] == 1, y’[0] == 0}, y[x], x] 1) Take the Laplace Transform of the following functions ( a ) cos(3 t ) e 5 t (1) ( b ) δ ( t - 3) (2) ( c ) 2 H ( t - 3) e t (3) ( d ) 2 s + 1 s ( s + 1) (as done in class) . (4) Comment on the shifting theorems for a and c . 2) Take the inverse Laplace Transform of the following functions ( a ) 4 s s 2 + 9 (5) ( b ) e 21 s s ( s 2 + 16) (6) Comment on the shifting theorems for b . 3) Work with step functions. Here H ( t - a ) is a step function going from 0 to 1 at a . (a) Plot [ H ( t - 2) - H ( t - 4)] from t = 0 to t
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Unformatted text preview: = 6. (b) Plot cos 2 t from t = 0 to t = 6. (c) Plot [ H ( t-2)-H ( t-4)] cos2 t from t = 0 to t = 6. Think about what the step functions are doing. 4) Solve the following ODE, Frst using DSolve and then by taking the Laplace transform followed by the inverse Laplace Transform. y -5 y + 6 y = e t ; y (0) = 0; y (0) = 2 (7) 5) Invert F ( s ) = 2 s 2 + 5 1 s 3 (8) in two ways: irst, by using InverseLaplaceTransform directly. Second, by calculating the convolution integral. Calculate the inverse transforms of F ( s ) = 2 / ( s 2 + 5) and G ( s ) = 1 /s 3 individually, then calculate i t f ( t- ) g ( ) d. (9) Verify that you get the same answer....
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