30 review exam 3

30 review exam 3 - 18.03 Lecture#30 Nov 16 2009 notes...

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18.03 Lecture #30 Nov. 16, 2009: notes Function Laplace transform f F ( s ) = integraltext 0 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
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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 0 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. So this is about the forced damped oscillator x ′′ + 8 x + 20 x = f, x (0) = x (0) = 0 , (Forced damped oscillator) The forcing is being done by a careless graduate student, who initially applies a sinusoidal forcing of period 2 π , and only later realizes that he needs to change it to period π : f ( t ) = 0
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