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First-Order Equations: don't understand a simplification step

  1. Feb 5, 2015 #1
    There is a specific step when solving first-order equations introduced in my textbook when describing the process for solving with an integrating factor, and it has been popping up everywhere - I do not understand how it works and the book doesn't explain it well.

    Example 1:

    x(dy/dx) + y = sin(x) / x


    d/dx(xy) = sin(x) / x

    Example 2:

    e^-x (dy/dx) - y(e^-x) = e^(-4x/3)


    d/dx(y(e^-x)) = e^(-4x/3)

    I'm having trouble rationalizing how this step is performed. Thanks for any help!
  2. jcsd
  3. Feb 5, 2015 #2

    Simon Bridge

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  4. Feb 6, 2015 #3


    Staff: Mentor

    In both examples they are recognizing that the left side is the derivative, with respect to x, of a product. In the first example, the product is xy. In the second, the product is ye-x.
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