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B First Order Non-Linear ODE (what method to use?)

  1. Feb 14, 2017 #1

    The problem is to solve:
    dy/dx = −[2x + ln(y)]*(y/x)

    I have tried to see if it is exact, I found it not to be, I can't easily find a function to multiply by to make it exact either (unless I am missing something obvious). It clearly isn't seperable, nor is it homogenous (I know although some non-homogenous ODEs can be solved using the homogenous method I don't think this one can be), it also isn't a linear equation so we can't use integrating factors.

    I'm looking for pointers in the right direction/which method to use.

    Thanks for your time
  2. jcsd
  3. Feb 14, 2017 #2


    Staff: Mentor

    Have you tried guessing a solution?

    Something like ##a^x## or something that works well with the natural log function.
  4. Feb 15, 2017 #3


    Staff: Mentor

    Did you figure it out?
  5. Feb 17, 2017 #4
    I think you are going the right way; checking the exact form, homogeneity etc.
    Here since there is a logarithmic function, I recommend substituting y=et where t is parameter and then be solved by finding integrating factor.
  6. Feb 22, 2017 #5
    Using y = exp(qx), where q is a function of x , I got it to separate, giving
    q = (A/x2) - 1, where A is a constant of integration.
    It seems to check.
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