6.2 - Math 334 Lecture #24 6.2: Solving IVPs with the...

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Unformatted text preview: Math 334 Lecture #24 6.2: Solving IVPs with the Laplace Transform Some Properties of the Laplace Transform. (1) The Laplace transform is linear: L{ c 1 f 1 ( t ) + c 2 f 2 ( t ) } = Z e- st c 1 f 1 ( t ) + c 2 f 2 ( t ) dt = lim A Z A e- st c 1 f 1 ( t ) + c 2 f 2 ( t ) dt = lim A c 1 Z A e- st f 1 ( t ) dt + c 2 Z A e- st f 2 ( t ) dt = c 1 lim A Z A e- st f 1 ( t ) dt + c 2 lim A Z A e- st f 2 ( t ) dt = c 1 Z e- st f 1 ( t ) dt + c 2 Z e- st f 2 ( t ) dt = c 1 L{ f 1 ( t ) } + c 2 L{ f 2 ( t ) } . (2) On continuous functions, the Laplace transform is 1- 1: L{ f ( t ) } = L{ g ( t ) } f ( t ) = g ( t ) . In particular, the only continuous function f ( t ) for which L{ f ( t ) } = 0 is f ( t ) = 0. (3) The Laplace transform is invertible: L{ f ( t ) } = F ( s ) L- 1 { F ( s ) } = f ( t ) . (4) The inverse Laplace Transform, L- 1 , is linear: L- 1 { c 1 F 1 ( s ) + c 2 F 2 ( s ) } = c 1 L- 1 { F 1 ( s ) } + c 2 L- 1 { F 2 ( s ) } ....
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6.2 - Math 334 Lecture #24 6.2: Solving IVPs with the...

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