This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: ODEs - Review Sheet 3 More on the Laplace transform Impulses and their Laplace transforms If F is a force and it acts during a time interval [ a,b ], then the impulse transmitted by the force is R b a F ( t ) dt. A unit impulse is one for which R b a F ( t ) dt = 1 . And an instantaneous unit impulse is an idealization in which the force is transmitted at a single instant of time. We write the idealized force as ( t- t ) (the Dirac delta function). It has the following properties: 1. R - f ( t ) ( t- t ) dt = f ( t ). 2. L ( ( t- t )) = e- st . (So in particular, L ( ( t )) = 1.) 3. L ( f ( t ) ( t- t )) = f ( t ) e- st . Note: The second and third properties follow immediately from the first. Convolutions: The convolution of f and g is the function ( f * g )( t ) given by ( f * g )( t ) = Z t f ( t- ) g ( ) d = Z t f ( ) g ( t- ) d. The importance of the convolution here is due to the following result: L ( f * g ) = L ( f ) L ( g ) , or L- 1 ( FG ) = L- 1 ( F ) * L- 1 ( G ) . This means that, for the ODE y 00 + by + cy = ( t ) , y (0) = y (0) = 0 , Y ( s ) = 1 s 2 + bs + c This special Y , which is the reciprocal of the characteristic polynomial, is denoted...
View Full Document