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    Differential Equations and Determinants

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    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!

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    https://brainmass.com/math/calculus-and-analysis/differential-equations-and-determinants-134378

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    ===== Q.1======= Response from OTA ================

    Determinant method means solving a matrix or inverting a matrix..same as solving

    simultaneous equations..but in this case, you can easily eliminate C1 and C2 as

    follow:

    I will use "^" for the exponent sign and use

    an easier notation: call (dy/dx) = D; (d^2y/dx^2) = D^2

    Let us say we want to find ...

    Solution Summary

    Differential equations and determinants are investigated.

    $2.49

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