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11/08/11 - Properties of Laplace Transforms and Inverting Laplace Transforms
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 | Basic Properties of the Laplace Transform
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| Laplace Transform of a function
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| Laplace Transform of a first-derivative
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Use integration by parts:
Simplify:
The transform of a derivative becomes a multiplication of the transform by s, minus the initial position y(0).
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| Laplace Transform of a second-derivative
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Use integration by parts:
Last integral is sL{y'(t)}. Substitute and simplify:
The transform of the second derivative becomes a multiplication by s-squared, minus s times the initial position y(0), minus the initial velocity y'(0).
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| Laplace Transform of a function multiplied by an exponential
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The transform of a function multiplied by an exponential shifts the s-variable in the transform domain.
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| Derivative of a Laplace Transform
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The first derivative of the Laplace Transform of a function is the transform of (-t•y(t)).
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| Each additional derivative adds another factor of -t, so:
The n-th derivative of the transform of a function is the transform of the function multiplied by (-t)n.
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| Inverting Laplace Transforms
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| Problem 5.4.18 by hand (with tables)
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| 5.4.18 - ODE with initial conditions:
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| Laplace Transform of ODE:
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| Invert Laplace Transform:
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| Read Section 5.2, pp. 197-201.
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| Do exercises p. 202 (25,29,33,41,44).
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| Read Section 5.3, pp. 203-208.
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| Do exercises p. 208 (1,7,11,15,19,25,31,35).
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| Read Section 5.4, pp. 209-214 of the text.
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| Do exercises p. 214 (5,19,27,31,32,35).
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