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27 laplace II solving ODEs

# 27 laplace II solving ODEs - 18.03 Lecture#27 Nov 9 2009...

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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; it’s your job to remember it! I’ll 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 don’t 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 0 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 ) > 0 f sF ( s ) f (0) Re( s ) > C f exponential type C tf F ( s ) Re( s ) > C f exponential type C u 0 ( t a ) f ( t a ) e as F ( s ) Re( s ) > C a > 0, f exp type C δ a e as all s a 0 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. Remember that u 0 ( t ) = braceleftbigg 0 t < 0 , 1 t > 0 . is the Heaviside unit step function; u 0 ( t a ) is what we called u a . Also the generalized function δ a ( t ) = Dirac delta function at a is the (generalized) derivative of u a . Remember also that to say f has exponential type C

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27 laplace II solving ODEs - 18.03 Lecture#27 Nov 9 2009...

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