Math 135 Homework, Winter 2015

Below are the weekly homework assignments. Nine of these will be collected in class on the due date. They may be turned in early, but not late. These each will be graded on a scale of 20 points. Your six best scores will be applied to your total homework score. The uncollected assignments cover material that will be tested on subsequent exams, so should be studied.

Homework Assignments

1. due Friday, 9 January
   • Section 14: 7, 8, 9, 10.
     Problem 7 should read x^2 y'' - 3 x y' - 5 y = 0 .
   • Section 15: 2, 3, 6a, 8, 9.
2. due Friday, 16 January
   • The characteristic polynomial of a differential operator L has roots:
     -1 + i2, -1 + i2, -1 - i2, -1 - i2, 5, 5, 5, 0, 0.
     
     a. What is the order of L?
     b. Give a general solution of Ly=0.
   • Find the Green function for the differential operator L = D^3 + 9 D .
   • Section 17: 2a, 2b, 2c, 8.
   • Section 18: 1a, 1c, 1f, 2.
3. due Friday, 23 January
   • Use the definition of the Laplace transform to compute L[u(t - 3) e^{2t}](s) ,
     where u is the unit step function.
   • Let f(t) and g(t) respectively have exponential orders a and b as t tends to infinity.
     a. Show that f(t) + g(t) has exponential order max{ a , b } as t tends to infinity.
     b. Show that f(t) g(t) has exponential order a + b as t tends to infinity.
   • Section 48: 2, 3, 4b, 4d.
   • Section 49: 2c, 3, 4.
4. due Friday, 30 January
   • Let f(t) be the list function given by
     • t^2 for 0 <= t < 3 ,
     • 12 - t for 3 <= t < 7 ,
     • 5 for 7 <= t ,
     where <= means "less than or equal to".
     Compute its Laplace transform L[f](s) .
   • Solve the initial-value problem
     y'' - 3 y' - 4 y = f(t) , y(0) = 2 , y'(0) = 0 ,
   where f(t) is the list function in the previous problem.
   • Section 50: 2b, 2c, 3b, 3d, 3e.
   • Section 51: 2a, 3b, 8a, 8b.
   • Section 52: 1, 2b, 5.
5. Friday, 6 February (not collected)
   • Section 68: 1, 3b.
   • Section 69: 2a, 2b, 6.
   • Sample Problems for the Midterm Exam - posted under "Exams".
6. due Friday, 13 February
   • Compute the Fourier series for the Euler "plucked string" given by
     • f(x) = - pi - x for x in [-pi,-pi/2) ,
     • f(x) = x for x in [-pi/2,pi/2] ,
     • f(x) = pi - x for x in (pi/2,pi] .
   • Compute the Fourier series for the Fourier "hot block" given by
     • f(x) = - 1 for x in [-pi,-pi/2) ,
     • f(x) = 1 for x in [-pi/2,pi/2] ,
     • f(x) = - 1 for x in (pi/2,pi] .
   • With software of your choice, for each Fourier series found in the
     previous two problems plot on the same graph the three functions:
     • the partial sums that contains the first four nonzero terms,
     • the partial sums that contains the first eight nonzero terms,
     • the function f(x) .
     There should be two graphs (one for each series), each with three
     functions plotted on it over the interval [-pi,pi] .
     What differences do you see in the two graphs?
   • Section 35: 1, 2, 3, 11a, 11b.
   • Section 36: 2a, 2b, 3.
7. due Friday, 20 February
   • Section 38: 1, 2, 5, 7.
   • Section 37: 2, 3, 4, 5.
8. due Friday, 27 February
   • Section 34: 5, 6, 7.
     Note: A function satisfies the "Dirichlet conditions" if it
     • is defined and bounded over [-pi,pi),
     • has only a finite number of discontinuities in [-pi,pi),
     • has only a finite number of local extrema in [-pi,pi).
   • Section 40: 1, 3, 5, 7.
   • Section 41: 2, 3, 4, 5.
9. due Friday, 6 March
   • Section 42: 1, 3, 4.
   • Section 43: 1 - 4, 6 - 9.
10. due Friday, 13 March
    • Section 66: 2, 3, 4, 5, 7.
    • Section 67: 2, 3, 4, 6.
11. due Thursday, 19 March (not collected)
    • Sample Problems for the Final Exam - posted under "Exams".

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