L17E - Some Laplace Transforms: f ( t) tn cos t sin t Cosht...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Some Laplace Transforms: f ( t ) F ( s ) t n n ! /s n +1 cos t s/ ( s 2 + 1) sin t 1 / ( s 2 + 1) Cosh t s/ ( s 2 - 1) Sinh t 1 / ( s 2 - 1) e ct f ( t ) F ( s - c ) f ( ct ) F ( s/c c f 0 ( t ) sF ( s ) - f (0) f 00 ( t ) s 2 F ( s ) - sf (0) - f 0 (0) u c ( t ) f ( t - c ) e - cs F ( s ) δ c ( t ) e - cs ( f * g ) ( t ) F ( s ) · G ( s )
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
The Method To solve the CCSOLIVP ay 00 + by 0 + cy = g ( t ) , y (0) = y 0 , y 0 (0) = y 0 0 we compute the Laplace transform Y ( s ) of y ( t ) : Y ( s ) = asy 0 + ay 0 0 + by 0 as 2 + bs + c + G ( s ) as 2 + bs + c whence y ( t ) = L - 1 ± asy 0 + ay 0 0 + by 0 as 2 + bs + c ² | {z } y h ( t ) + L - 1 ± G ( s ) as 2 + bs + c ² | {z } y p ( t ) Clearly y h is the solution of the homoge- neous CCIVP ay 00 + by 0 + cy = 0 , y (0) = y 0 ,y 0 (0) = y 0 0 . ( H ) 2
Background image of page 2
and y p ( t ) is the convolution of g ( t ) with the solution of the homogeneous CCIVP ay 00 + by 0 + cy = 0 , y (0) = 0 ,y 0 (0) = 1 /a . 3
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Examples Example: Find the Laplace transform of the saw function g ( t ) def = X n =0 u nc ( t ) ( t - nc ) - u ( n +1) c ( t ) ( t - ( n +1) c - 1 ) 4
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 09/07/2010 for the course PHY 303L taught by Professor Turner during the Spring '08 term at University of Texas at Austin.

Page1 / 13

L17E - Some Laplace Transforms: f ( t) tn cos t sin t Cosht...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online