ps4final

# ps4final - 18.03 Problem Set 4 Due by 12:45 P.M Friday in...

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Due by 12:45 P.M., Friday, March 14, 2008, in the boxes at 2-106, next to the Under- graduate Mathematics Oﬃce. I encourage collaboration on homework in this course. However, if you do your homework in a group, be sure it works to your advantage rather than against you. Good grades for homework you have not thought through will translate to poor grades on exams. You must turn in your own writeups of all problems, and, if you do collaborate, you must write on the front of your solution sheet the names of the students you worked with. Because the solutions will be available immediately after the problem sets are due, no extensions will be possible . L13 W 5 Mar Driven systems: Superposition, sinusoidal harmonic response: Notes O.1; EP 2.6, pp. 157–159 only—see SN 7 if you want to learn about beats. R9 Th 6 Mar Complex numbers and phase lag. L14 F 7 Mar Operators, exponential response formula: SN 10, Notes O.1, 2, 4, EP 2.6 (pp. 165–167). L15 M 10 Mar Undetermined coeﬃcients; RLC circuits: SN 11, EP 2.5 (pp. 144–153); EC 2.7, SN 8. R10 T 11 Mar Operators, ERF, undetermined coeﬃcients. L16 W 12 Mar Frequency response: SN 14. R11 Th 13 Mar Ditto. L17 F 14 Mar Linear time invariant operators. Part I. 13. (W 5 Mar) (a) [From R8.] Two facts about solutions to m ¨ x + b ˙ x + kx = 0: (i) The real part of a solution is again a solution. (ii) Any constant multiple of a solution is again a solution. So if e rt is a solution (with r = a + ci a complex root, say), then so is z 0 e rt , for any complex number z 0 . If we write z 0 = Ae - φi (so A = | z 0 | and φ = - Arg ( z 0 )), what is the real part of z 0 e rt ? Explain why this gives the general solution. What would you take for z 0 to get Im e rt ? (b) For each of the following functions f ( t ), ﬁnd a complex number z 0 and a positive real number ω such that f ( t ) = Re ( z 0 e iωt ). (i) cos(2 t ). (ii) 2 sin( πt ). (iii) cos( t + π 4 ). (iv) cos( t ) + 3 sin( t ). (c)

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## This note was uploaded on 02/06/2009 for the course 18 18.03 taught by Professor Unknown during the Spring '09 term at MIT.

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ps4final - 18.03 Problem Set 4 Due by 12:45 P.M Friday in...

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