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Math Help - Eliminating the arbitrary constant

  1. #1
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    Eliminating the arbitrary constant

    Hi,

    I am having trouble understanding the steps needed to eliminate the arbitrary constants from the equation y = Ax + 3. The answer in the book is y = x (dy/dx) + 3. Here are my steps along with my reasoning for each step (please correct my reasoning so I can finally understand how to do this):

    y = Ax + 3

    dy/dx = A (I am still left with A, so I need to find another way to get rid of A)

    y = Ax + 3 (starting over)

    (y - 3)/x = A (isolate A)

    (y - 3)/x = dy/dx (replace A with dy/dx - is this correct? if so, why?)

    y - 3 = (dy/dx)x

    y = (dy/dx)x + 3 (isolate y. Now we are left with an equation that does not have A in it )
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  2. #2
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    Solve for A, differentiate both sides using implicit differentiation. HT: Danny.
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by sparky View Post
    Hi,

    I am having trouble understanding the steps needed to eliminate the arbitrary constants from the equation y = Ax + 3. The answer in the book is y = x (dy/dx) + 3. Here are my steps along with my reasoning for each step (please correct my reasoning so I can finally understand how to do this):

    y = Ax + 3

    dy/dx = A (I am still left with A, so I need to find another way to get rid of A)

    y = Ax + 3 (starting over)

    (y - 3)/x = A (isolate A)

    (y - 3)/x = dy/dx (replace A with dy/dx - is this correct? if so, why?)

    y - 3 = (dy/dx)x

    y = (dy/dx)x + 3 (isolate y. Now we are left with an equation that does not have A in it )
    Am I missing some detail here?
    y = Ax + 3

    dy/dx = A

    Put this value of A into the original equation:
    y = (dy/dx)x + 3

    which is the desired solution.

    -Dan
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  4. #4
    A Plied Mathematician
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    Quote Originally Posted by topsquark View Post
    Am I missing some detail here?
    y = Ax + 3

    dy/dx = A

    Put this value of A into the original equation:
    y = (dy/dx)x + 3

    which is the desired solution.

    -Dan
    Certainly nothing wrong with this method, and it is the most direct. However, it is not very general. Danny's method will work in pretty much any case that has a single arbitrary constant for which you can solve (which is true for many first-order ODE's).
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