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Exact equations

  1. Sep 17, 2010 #1
    I have the following exact equations, however, teacher said it is incorrect. Cannot find a mistake. Could you please help me?

    (3x^2 - 2x - y) dx + (2y - x + 3y^2)dy = 0

    This is exact equation, because:
    P = (3x^2 - 2x - y)
    Q = (2y - x + 3y^2)
    P'y = Q'x = -1

    Then integrate Intx P = x^3 - x^2 - yx + fi(y).

    Find derivative: (x^3 - x^2 - yx + fi(y)'y = -x+ fi'(y) = Q = (2y - x + 3y^2).
    Find fi'(y) = 2y + 3y^2. Iegūst fi(y) = y^3 + y^2.

    Result: x^3 - x^2 - yx + y^3 + y^2 + C.


    Where is a mistake? Mabybe the whole idea is incorrect? Please, help me.
     
    Last edited: Sep 17, 2010
  2. jcsd
  3. Sep 17, 2010 #2
    I don't see any mistakes.
     
  4. Sep 17, 2010 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    A better notation of "the derivative with respect to y" than ( )'y is ()_y. The former is too likely to be confused with "the derivative times y".

    The mistake is that you have an expression but no function! You need to write
    [itex]x^2- x^2- yx+ y^3+ y^2= C[/itex]
    or
    [itex]x^2- x^2- yx+ y^3+ y^2+ C= 0[/itex]
    (or any other constant on the right side.)
    Those are now "implicit functions" that could, theoretically, be solve for x or y. Just the expression [itex]x^3- x^2- yx+ y^3+ y^2+ C[/itex] is not. Your teacher is being very strict but you need to learn to be precise in mathematics.
     
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