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Unformatted text preview: 18.03 Lecture #27 Nov. 9, 2009: notes Today we talk about solving ODEs using Laplace transform. Since Laplace transforms are defined only for functions that are zero for t < 0, we can only discuss solutions for t 0. This point is treated sloppily in the book and in most sources; its your job to remember it! Ill start with a chart recording some of the main facts you need to know about Laplace transforms. (The one I wrote on the board in class, and the one in the preliminary version of these notes, was missing a factor of (s-a) in the formula for t n e at .) There will be charts like this on Exam 3 and on the final, so you dont really need to memorize all the formulas; but you need to know what they mean and how to use them. Some commonly used notation: a function of t is written with a lower case Roman letter (like f , g , x , or y ). The Laplace transform is written with the corresponding upper case Roman letter: F ( s ) = L ( f )( s ) , G ( s ) = L ( g )( s ) , X ( s ) = L ( x )( s ) , Y ( s ) = L ( y )( s ) . Function Laplace transform range of definition notes f F ( s ) = integraltext f ( t ) e ts dt Re( s ) > C f exponential type C af + bg aF + bG Re( s ) > max( C f ,C g ) f , g exp types C f , C g e at 1 s a Re( s ) > Re( a ) t n n ! s n +1 Re( s ) > f sF ( s ) f (0) Re( s ) > C f exponential type C tf F ( s ) Re( s ) > C f exponential type C u ( t a ) f ( t a ) e as F ( s ) Re( s ) > C a > 0, f exp type C a e as all s a e at cos( bt ) s a ( s a ) 2 + b 2 Re( s ) > a e at sin( bt ) b ( s a ) 2 + b 2 Re( s ) > a t n e at n ! ( s a ) n +1 Re( s ) > Re( a ) f ( n ) s n F ( s ) s n 1 f (0) f ( n 1) (0) Re( s ) > C f ( n ) exponential type C t n f ( 1) n F ( n ) ( s ) Re( s ) > C f exponential type C Table of Laplace transforms....
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This note was uploaded on 05/06/2010 for the course 18 18.03 taught by Professor Unknown during the Fall '09 term at MIT.
- Fall '09