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

Singular solutions Separable Eq. and IVP

  1. Feb 19, 2009 #1
    Hi,

    I am a little bit confused about singular solutions and their relationship with IVP's and decided to ask you.


    As far as I understood, the IVP's could be in a form:
    y' = f(x,y)
    y(x0) = y0

    To obtain a general solution, we could use separable eq. method. I have learned that sometimes when we separate the variables, we assume a particular condition to be valid. (that might have an asymptotic behaviour. )

    Also, as far as I know, a singular solution can not be obtained from a general solution.

    My impression was that the general solution shouldn't have worked for y = y0 if we derived it assuming y is not equal to y0. However, that example confuses me;

    y' = 2 (y^1/2)
    y(0) = 0


    By the sep. eq. method and integrating both sides;

    (dy / 2(y^1/2)) = dx

    I have found that y = (x-c)^2. However, while separating, we actually assume that y is not equal to 0.

    What's confusing is, at this general solution, when I plug y = 0 and x = 0; I get a particular solution; which is y = x^2 and that perfectly works well! It does not make sense to get a particular solution which suggests y= 0 when x = 0 using a general solution that is not defined for y = 0. (Because I had assumed y is not equal to 0 while I was separating the variables. )
     
    Last edited: Feb 19, 2009
  2. jcsd
  3. Feb 19, 2009 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    It is sufficient to solve for y with y not 0, then extend the solution to y= 0 since, in order that the derivative exist, y must be continuous.

    But did you notice that y(x)= 0 for all x is also a solution? In fact, if you set y(x)= (x- a)2 for x< a< 0, y(x)= 0 for a< x< b (where b> 0), y(x)= (x-b)2 for 0< b< x is also a solution to the differential equation satisfying y(0)= 0.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Singular solutions Separable Eq. and IVP
  1. Singular Solution (Replies: 3)

  2. Singular Solutions (Replies: 4)

Loading...