Mathematics 427K          Assignments from 3rd exam to Final
     
  • Lecture 24 :  4/13 :     Series Solutions of Ordinary Differential Equations

    • Read sections 5.1, 5.2  and 5.3(it does not hurt to review calculus! )Suggested Problems:
    page 231, # 17,18; page 241; #1-4, page 247; #1-4.
  • Lecture 25 :   4/18:     The guituar string, harmonics, pitch and the musical scale:

    • Read section 10.6.  Start on problems on separation of variables.
  • Lecture 26 :  4/20    Separation of Variables and Fourier Series,

    • Sections 10.1 and 10.2, Page 550-551 4, 7-12; Page 561 #13-18.
  • Lecture 27:  4/25  More on Fourier Series, Section 10.3, page 567, # 1-12.
  • Lecture 28: 4/27   Even and Odd Functions, section 10.4, page 575, # 1-12.
  • Lecture 29:  5/2  More on partial differential equations  Sections 10.5 and 10.6 (again), page 587 # 1-4, page 597 #1-5.
  • Lecture 30:5/4  Review for Final; review of practice final.

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  • There are some very interesting facts that it is not so easy to know from simple math. The one classy fact

    • we have obtained so far is that:                i(pi)
                                                                   e        = -1.
  • Lets look at the Fourier series for the square wave:

    •                for,   0  <- 1 < x < 0 ,    f(x) = - 1,
                   for   0 < x < 1,       f(x) = 1.
    Extend the function  to be of period two on the entire line. This gives a square square wave of period 2.
    There are questions about this on your computer homework. The Fourier series for this square wave is

                       f(x) =   1/2  +   2/(pi)[ sin((pi)x)  + 1/3 sin (3 (pi) x ) + 1/5 sin (5 (pi) x ) +
    1/7 sin (7(pi)x)  +   1/9 sin(9(pi)x) +   ......]     =   1/2 +    2/(pi)   (sum  ) sin((2n-1)(pi)x).

    This is worked out on page 565 and 566 (set L = 1). What do you get if you set x = 1/2? Do you know any way to check this? See problem 35 on page 576.

  • 2. The following was discovered in l674 (see problem 36 on page 576).

    •                        2                                                                 infinity
                      (pi)/8 = 1 + 1/9 + 1/25 + 1/49 + 1/81 ... =       sum    [ 1/2n+1)].
                                                                                                n =0
    Check this on your calculator. Problem 36 shows how to get this from the Fourier series for the triangular wave (shown in lecture).