(Issued Wed, Nov. 22)

Due Fri, Dec 1

(Note: I will assign the next homework (#14) on Wed, Nov 29, i.e. back on schedule. But since this one is given over the holiday, you have an extra 2 days to turn it in...)

Reading for this assignment: Boas Ch 8.1-8.3, 8.5, 8.6 (start), 12.1-12.2

1 Boas 8.1.4 Hint:

2a) 8.2.5 2b) 8.2.7

Hint: Boas section 8.2 is titled "separable equations".

3) Boas 8.2.12

4) Boas 8.2.19a

(This type of problem is quite common in physics! Look at Boas problems 19b or 20 or 21 if you'd like to see other different-looking problems that are mathematically equivalent)

5a) Boas 8.5.24 5b) Boas 8.5.31.c only.

Hint for a) You will have to recall from our study of complex numbers how to find cube roots!

Hint for b) To prove such an identity, you must remember that D is an operator. So, the given equations really should have a "y(x)" tacked on at the right of both sides. (See the description Boas gives at the start of the problem, where she proves the identity D(D+x)=D^2+xD+1. But, what she really showed is that D(D+x)y = D^2y+xDy+y. If it's an identity, then the equation must hold no matter what y(x) is! )

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You do not have to do any Mathematica this week; in fact, I prefer that you work out these problems analytically. But you might like to know that Mma can solve (check?!) many of these problems for you. The command is DSolve[...], a symbolic differential equation solver! The syntax required is rather specific. Blachman p. 59, In[51] shows a simple example, and In[52] shows an example in which she specifies a "boundary condition". (Note (!) that these equations all use double equals, "==")

In many cases, differential equations can't be solved analytically (i.e., there's no way of writing down y as an elementary function), but there's still a solution, you just can't write down a simple formula for it! Mma can find this as well, with the NDSolve[...] command, which is a "numerical differential equation solver". This requires that you provide a suitable number of boundary conditions, and results in what is known as an interpolating function, which you can plot. See Blachman p. 39 and 40 for an example of this. (I find this feature of Mma pretty useful - maybe you will someday as well!)

If you like, here is an extra credit problem which will demonstrate the use of DSolve and NDSolve.