Beginning Differential Equation & Pedantic Diff Calculus Questions
(2 total)
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To whom it may concern:


First and foremost, greetings from Brooklyn, NY, USA!

Although I'm a part-time adult college student, my question does not related to any class I am currently taking. I'm merely trying to sharpen an improve my rusty (and never particularly sharp) math skills.

Just for fun, I began reading my "new" book, "Differential Equations" by E. J. Maurus, published in 1917 (perhaps "new" was the wrong word ;-). I've run into two questions already (and I'm only on page two!); perhaps you can take a shot at them?

N.B. Monospace font required.


Question 1:
-----------

Quoting from the book:
-----------------------------------------------------------------
Example 2. Form a differential equation by eliminating the constants c1 and c2 from the equation

              x        2x
      y = c1*e  +  c2*e

Since there are two constants to eliminate, three equations are needed. The two extra equations are formed by differentiating the given equation twice, giving
                                        2
      dy       x         2x            d y       x         2x
      -- = c1*e  + 2*c2*e       and    --- = c1*e  + 4*c2*e
      dx                                 2
                                       dx

Eliminating, by determinants or otherwise, we have
                     2
                    d y     dy
                    --- - 3 -- + 2y = 0
                      2     dx
                    dx
-----------------------------------------------------------------
Quotation ends.


I am perfectly comfortable (okay, at least reasonably comfortable ;-) with differential equation related parts of the problem; it's the "Eliminating, by determinants or otherwise" part that stymies me. How would I employ determinants (with which I have some familiarity) to obtain the above answer, and what are some of the other methods I could have used? (I never managed to obtain the final result, although I believe I did verify it.)



Question 2:
-----------

In addition to having an interest, but no particular facility, in math, I am similarly predisposed towards (English) syntax (grammar). Perhaps this explains why my curiosity was piqued by the following question:


[In the interest of brevity, I shall edit more heavily this time.]

Quoting from the book:
-----------------------------------------------------------------
Example 1. Form a differential equation by eliminating the constant c from the equation x*y = c*sin(x).

...

...differentiating the given equation: x dy + y dx = c * cos(x) dx

Eliminating the c, we have the differential equation
       dy
     x -- + y = x*y*cot(x)
       dx
-----------------------------------------------------------------
Quotation ends.


Once again, it is not the Diff Equation solving that I have a question with. It is the odd (to me) syntax of:

	x dy + y dx = c * cos(x) dx

Of course I threw away my scrap paper, but I think I (eventually) solved the problem as follows:

                                  x*y
     x*y = c*sin(x)    -->   c = -----
                                 sin(x)

     Dx[x*y = c*sin(x)]

     Dx[x*y] = Dx[c*sin(x)]

           (by chain rule)
     +-                     -+
     |  d(x)            d(y) |
     |  ---- * y +  x * ---- | = c * cos(x)
     |   dx              dx  |
     +-                     -+

     +-               -+
     |            d(y) |
     |   y +  x * ---- | = c * cos(x)
     |             dx  |
     +-               -+

     +-               -+
     |            d(y) |     x*y
     |   y +  x * ---- | = ------ * cos(x)
     |             dx  |   sin(x)
     +-               -+


     +-             -+
     |      d(y)     |   cos(x)
     |  x * ---- + y | = ------ * x*y
     |       dx      |   sin(x)
     +-             -+


            d(y)
        x * ---- + y  = cot(x) * x*y
             dx
                                             QED

You'll observe that at no time did I write something like:
    "x dy + y dx = c * cos(x) dx"
(Or did I?)

Mr. Maurus was not a man to mince words; his entire "Differential Equations" book is less than 7in x 5in and numbers only about fifty pages. If he wrote "x dy + y dx = c * cos(x) dx", then I'm sure he knew what he was doing. Can you explain his terminology, or point me to a math book, web page, etc. that does?

Thanks in advance for your help!


Cordially,
Richard Kanarek
RKanarek@Spamcop.net