M334F11ECP - Question 1 A function f defined on R is...

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Math 334: Optional Extra Credit Project Extra Credit Points Available: 1% Due Date: last day of classes, Dec. 8, 2011. No Exceptions. Select one of the following two extra credit projects and submit it by the deadline. (Submitting both options will not double your extra credit!) Option 1 . Using a second-order nonconstant coefficient linear homogeneous differential equation to generate orthogonal polynomials. Question 1. The Legendre equation is (1 - x 2 ) y - 2 xy + N ( N + 1) y = 0 where N is a nonnegative integer. If y = n =0 a n x n , find the recurrence relation for a n . Question 2. For each N , show that the Legendre equation has a polynomial solution P N ( x ) of degree N . Question 3. Find the polynomials P N ( x ) for N = 0 , 1 , 2 , 3. Question 4. Show for any N that P N satisfies [(1 - x 2 ) P N ] + N ( N + 1) P N = 0. Question 5. An inner product on C [ - 1 , 1], the vector space of functions continuous on [ - 1 , 1], is f, g = 1 - 1 f ( x ) g ( x ) dx. Show that P M , P N = 0 when M = N . Hint: start with the equation in Question #4. Option 2 . Solving a periodically forced mass-spring system where the periodic forcing
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Unformatted text preview: Question 1. A function f defined on R is periodic with period T > 0 if f ( t + T ) = f ( t ) for all t in R . Sketch the graph of the saw-tooth function g ( t ) = t , 0 ≤ t < 1, g ( t +1) = g ( t ), over the interval [0 , 4]. Question 2. Find the Laplace transform of a periodic function f ( t ) with period T . Hint: L{ f ( t ) } = lim A →∞ Z A e-st f ( t ) dt = Z T e-st f ( t ) dt + lim A →∞ Z A T e-st f ( t ) dt, and then make the change of variable t = ξ + T on the last integral. Question 3. Find the Laplace transform of the saw-tooth function g in Question #1. Question 4. Find the inverse Laplace transform of F ( s ) = 2 (1-e-πs )( s 2 + 1) . Hint: could you use the geometric series somehow? Question 5. Use the Laplace transform to solve the initial value problem u 00 + 3 u + 2 u = g ( t ) , u (0) = 0 , u (0) = 0 , where g is the saw-tooth function from Question #1....
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