Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Solution to nonlinear ODE with radicals

  1. Jul 19, 2012 #1
    I am not too familiar with differential equations but am familiar with basic calculus, I came across this equation trying to describe a particular function:
    dy/dx =((sqrt((y-x)^2+y^2)-abs(y))/(y-x))*abs(y)/y

    Anyway I tried to separate the variables unsuccessfully and using v(x)=y(x)/x with no success, I really tried and couldn't solve it, so some insight would be appreciated. I tried restricting y to be positive, still without a breakthrough. Sorry if it's confusing, can't figure out the math format.
     
  2. jcsd
  3. Jul 19, 2012 #2

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    There is a solution y = cx, if that helps. c satisfies a quartic. One of the roots is 1, but you can discount that, leaving a cubic. All the real roots of that are negative.
     
  4. Jul 19, 2012 #3
    Are you suggesting that I try to solve the differential equation with respect to c? Could you elaborate a little since I fail to see c satisfying a quartic.
     
  5. Jul 19, 2012 #4

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Suppose y > 0 in some domain:
    dy/dx = (√((y-x)2+y2)-y)/(y-x)
    Put y = cx:
    c = (√((c-1)2+c2)-c)/(c-1)
    c(c-1) + c = √((c-1)2+c2) = c2
    (c-1)2+c2 = c4
    (c-1)2 = c2(c2-1)
    There's a common factor c-1. That root corresponds to y=x, which makes the original equation indeterminate. Otherwise:
    (c-1) = c2(c+1)

    OTOH, where y < 0:
    dy/dx = -(√((y-x)2+y2)+y)/(y-x)
    Put y = cx:
    c = -(√((c-1)2+c2)+c)/(c-1)
    c(c-1) - c = -√((c-1)2+c2) = c2 -2c
    etc.
     
  6. Jul 19, 2012 #5
    Thanks for the reply, but I was looking for a more general solution where y is not necessarily a linear function of x, perhaps where c would also be a function of x. It seems that your solution forces the y to be zero when x is zero. If c is a function of x then,

    c+x*dc/dx =(√((c-1)2+c2)-c)/(c-1)
    x*dc/dx=(√((c-1)2+c2)-c^2)/(c-1)
    Then, I am quite at a loss trying to integrate and solve that for c
     
  7. Jul 19, 2012 #6

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Of course. I was merely pointing out one set of solutions. These should shed light on the overall behaviour. Other solutions cannot cross the line y = cx. Finding all the real roots for the two cases (y >/< 0) will carve up the plane into separate domains.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Solution to nonlinear ODE with radicals
  1. Nonlinear ODE (Replies: 2)

Loading...