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Solution to the diff. eqn. dy/y-a*dx/x = b, where a and b are constant

  1. Mar 24, 2014 #1
    It is very clear that the solution to the equation "dy/y-a*dx/x = 0" is y=C*x^a. However I cannot figure out the solution when I add the constant b to the other side. Any help would be greatly appreciated!
  2. jcsd
  3. Mar 24, 2014 #2


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    Integrate both sides of the equation dy/y - a dx/x = b .
  4. Mar 24, 2014 #3
    Thanks for the hint but, integrate b with respect to what?
  5. Mar 24, 2014 #4


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    Well, I assume y is a function of a parameter -say- p: y(p).
    The same for x.

    And I assume the right hand side must have been some b dp.
    No idea what the exact question could be, of course.

    If this comes from a book, you could provide the reference.

    So my assumption is: you are asked to solve

    dy/y - a dx/x = bdp

    You need anyway a d-'something' (differential) on the rhs.
    So usual, that I even did not see it was missing in your question.
  6. Mar 24, 2014 #5
    The equation is not from any book, I derived it for a particular problem that I have.

    y is a function of x, y(x) and a and b are two constants.

    I do not know if it has a solution. The thing is that if b=0, the solution is very simple, y=C*x^a, so I was wondering whether there is a closed form solution if b is non-zero but a constant.
  7. Mar 24, 2014 #6


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    The right hand side must be a differential, not a simple constant.
    For example, you could rewrite the original equation as:

    1/y dy/dx - a /x = 0

    Here you have a derivative on the left hand side, not a differential anymore.
    You can then generalize it to:

    1/y dy/dx - a /x = b

    And you can transform it to:

    dy/y - dx/x = b dx

    where "elements" or "differential" appear on both sides.
    It should be like that!
    Something supposed to be "as small as needed" on the left cannot be equal a given constant on the right.
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