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Differential of a y mixed with x

  1. Jun 14, 2017 #1
    1. The problem statement, all variables and given/known data
    Find dy of ##~xy^2+x^2y=4##

    2. Relevant equations
    Differential of a product:
    $$d(uv)=u\cdot dv+v\cdot du$$

    3. The attempt at a solution
    $$2xy~dy+y^2~dx+x^2~dy+2xy~dx=0$$
    $$x(2y+x)dy=-(y+2x)dx$$
     
  2. jcsd
  3. Jun 14, 2017 #2

    ehild

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    You miss a y on the RHS.
     
  4. Jun 14, 2017 #3
    $$x(2y+x)dy=-y(y+2x)dx$$
    $$\frac{dy}{y}=-\frac{y+2x}{2y+x}~\frac{dx}{x}$$
    Is there a meaning for ##~\frac{dy}{y}~##?
     
  5. Jun 14, 2017 #4

    WWGD

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    What type of meaning are you considering? You can see it as ##\frac{1}{y} dy ##.
     
  6. Jun 15, 2017 #5
    And what is ##~\frac{1}{y} dy~##? what do i do with it? ##~d(ln~y)=\frac{1}{y} dy##
    But the book teaches logs only later
     
    Last edited: Jun 15, 2017
  7. Jun 15, 2017 #6

    scottdave

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    I think to solve your original problem, you want to rearrange it so that you have (dy/dx) = {something}
     
  8. Jun 15, 2017 #6
    I have to find ##~\frac{dy}{dx}=f(x)~##, but i have ##~\frac{dy}{dx}=f(x,y)##
     
  9. Jun 15, 2017 #6

    WWGD

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    I would guess this is a case of implicit differentiation, where you assume y is a function of x. Is that in your book?
     
  10. Jun 15, 2017 #7
    Yes, y=f(x) only
     
  11. Jun 15, 2017 #8

    WWGD

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    Then differentiate the whole expression as a function of x, using the chain rule on y=y(x).
     
  12. Jun 15, 2017 #9
    $$2xy\frac{dy}{dx}+y^2+x^2\frac{dy}{dx}+2xy=0$$
    $$dy=-\frac{y^2+2xy}{x^2+2xy}~dx$$
     
  13. Jun 15, 2017 #10

    WWGD

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    Good, but no need to split the dy/dx. Leave it as a single unit and solve for it.
     
  14. Jun 15, 2017 #11
    $$\frac{dy}{dx}=-\frac{y^2+2xy}{x^2+2xy}$$
    What i can do, and i can't even that, is to integrate, but it's not the point and the book didn't teach it yet.
    What else can i do with a derivative which involves y?
    I need dy, that is what i was asked, and any dy=f(x)dx
    I need to express, find, what is y=y(x)
     
  15. Jun 15, 2017 #12

    WWGD

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    Sorry, then do split into dy and dx parts.I am not clear, just what is your goal here, to find y?
     
  16. Jun 15, 2017 #13
    I was asked to find dy
     
  17. Jun 15, 2017 #14

    WWGD

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    My apologies then, do separate dy and dx and solve for dy.
     
  18. Jun 15, 2017 #15
    Now i see that the book's answer is like mine, leaving y undevelopped
     
  19. Jun 15, 2017 #16

    WWGD

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    " undeveloped"? How so? What is the book's answer?
     
  20. Jun 15, 2017 #17
    Snap1.jpg
     
  21. Jun 15, 2017 #18

    WWGD

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  22. Jun 15, 2017 #19
    Thank you scottdave and WWGD
     
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