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Homework Help: Linear Differential equation problem

  1. Mar 29, 2016 #1
    1. The problem statement, all variables and given/known data
    Solution of the differential equation
    ##(\cos x )dy = y (\sin x - y) dx , 0 < x < \dfrac{\pi}{2} ## is

    2. Relevant equations

    3. The attempt at a solution
    Only separation of variables, homogenous and linear DEs are in the syllabus, therefore it must be one of those. It obviously isn't the former two, so it must be a linear DE. I have no idea how to convert this into a linear form, especially because of the ##y^2## term. Please help.
  2. jcsd
  3. Mar 29, 2016 #2


    Staff: Mentor

    It's not linear, due to the y2 term. It might be homogeneous, which is an ambiguous term that can mean two different kinds of diff. equations.
  4. Mar 29, 2016 #3
    I'm not sure what the other homogenous is, I'm talking about the one in which we get y/x terms, and substitute that as V.
    Okay, so how do I proceed? It probably involves substitutions, but I'm not sure what to substitute.
  5. Mar 29, 2016 #4


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    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    You can't express y' as a function of y/x, so the differential equation isn't homogeneous. You need a different approach.

    You can rearrange the terms slightly to get ##(y\sin x)dx - \cos x\,dy = y^2\,dx##. Why would you want to do this? It's because the lefthand side is exact—that is, if ##v = -y\cos x##, you have
    $$dv = \frac{\partial v}{\partial x}\,dx + \frac{\partial v}{\partial y}\,dy = (y\sin x)dx - \cos x\,dy.$$ So try out the substitution ##v = -y\cos x## and see what you get after you rewrite the original differential equation in terms of ##v## and ##x##.
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