HW11
Assigned:       THU 12 NOV 09
Due:                THU 19 NOV 09

Problem 1.   Boas Problem #1, Ch.12.19, page 603. (Hint: We actually do this in detail in the online PDF notes for class.)

Problem 2.   Boas Problems #5 & #6, Ch.12.20, page 604.

Problem 3.   Boas Problems #13 & #15, Ch.12.20, page 605.

Problem 4.   Boas Problem #12, Ch.12.21, page 606. (Hint: Plug the two proposed solutions into the given Diffy-Q and show they both solve it (to give zero) by taking the solutions' derivatives carefully with the chain rule. Then expand these two solutions in a Taylor series around 1/x (which you get by just taking the series for sine and cosine with x replaced with 1/x) and show that these two solutions do not have the form of a Frobenius series, which is given in Eq.11.1, by eyeballing it. You will see that instead of just one or two terms with powers of x in the denominator you will have an infinite number of terms with ever increasing powers of x in the denominator. This is called an isolated essential singularity and is not of the Frobenius type.

Problem 5.   Boas Problem #1, Ch.12.22, page 611. (Hint: This is working out the derivatives using D=d/dx and operating the D operator left to right and then some algebra. For example (D+x)y = Dy+xy = y'+xy, etc.)

Problem 6.   Boas Problem #2, Ch.12.22, page 611. (Hint: (D+x)y = Dy+xy = y'+xy = dy/dx+xy = 0 is of the form variables separable (see hint in book). That is solve for y and dy on one side and x and dx on the other side and integrate both sides and then Exp both sides. Don't forget the constant of integration. See Eq.8.2.6 on page 395 for an example. Check your answer in Mathematica using DSolve.)