30 review exam 3

30 review exam 3 - 18.03 Lecture#30 Nov 16 2009 notes Function Laplace transform f F s = integraltext ∞ f t e − ts dt u a t f t − a e − as

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 18.03 Lecture #30 Nov. 16, 2009: notes Function Laplace transform f F ( s ) = integraltext ∞ f ( t ) e − ts dt u a ( t ) f ( t − a ) e − as F ( s ) f ′ sF ( s ) − f (0) f ′′ s 2 F ( s ) − sf (0) − f ′ (0) f ( n ) s n F ( s ) − s n − 1 f (0) −···− f ( n − 1) (0) e at f ( t ) F ( s − a ) tf ( t ) − F ′ ( s ) t n f ( t ) ( − 1) n F ( n ) ( s ) u a ( t ) e − as /s δ a e − as 1 1 /s e at 1 / ( s − a ) t n n ! /s n +1 t n e at n ! / ( s − a ) n +1 cos( bt ) s s 2 + b 2 sin( bt ) b s 2 + b 2 e at cos( bt ) s − a ( s − a ) 2 + b 2 e at sin( bt ) b ( s − a ) 2 + b 2 Properties of Laplace transforms. TYPO in notes for Lecture 28: the second Laplace transform table was missing a factor of 1 /s in the transform of u a . Topic for today is a review before the exam. The big ideas: Big idea one: Fourier series and Laplace transforms (and power series expansions, not covered in this unit) each encode a function in a way that makes differentiation into a simple algebraic oper- 1 ation. (Approximately “multiplication by n ” for Fourier series, “multiplication by s ” for Laplace transform; details in the magic charts below.) Big idea two: Taking the Fourier series or Laplace transform of a differential equation gives a simple algebraic equation for the transform of the solution; solving this equation (or seeing that no solution exists) is often easy. Medium idea three: Fourier series make sense only for functions that are 2 L-periodic for some positive L . They can be used only to investigate 2 L-periodic solutions , and only when the functions in the ODE are 2 L-periodic. Laplace transforms make sense ony for functions that are zero for t < 0. They can be used only to investigate solutions that are zero for t < 0, and only when the functions in the ODE are zero for t < 0. Medium idea four: It’s good to think of a Fourier series of a periodic φ as just the collection of numbers (Fourier coefficients) a n ( φ ) and b n ( φ ). The formula φ ( t ) = a 2 + ∞ summationdisplay n =1 ( a n ( φ )cos( nπt/L ) + b n ( φ )sin( nπt/L )) tells how to recover φ from the Fourier series. In the same way, we think of the Laplace transform of f as a function F ( s ). This time it’s more difficult to recover f from F , so there is some serious emphasis on techniques for doing that. I’ll start with a Laplace transform example. The plan is to include also a Fourier series example, but if I don’t get to it in lecture, maybe that one will be easier just to read here....
View Full Document

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.

Page1 / 6

30 review exam 3 - 18.03 Lecture#30 Nov 16 2009 notes Function Laplace transform f F s = integraltext ∞ f t e − ts dt u a t f t − a e − as

This preview shows document pages 1 - 3. Sign up to view the full document.

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