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I How to solve 2y+(y')^2+ax^b=0 ?

  1. Feb 26, 2016 #1
    I tried to solve this differential equation:

    2y+(y')^2+ax^b=0

    ...but don't know what to do with it.

    Don't know what variable substitution to use.
    Tried Taylor series, but I get horrible nonlinear equations for the coefficients.

    Tried Mathematica, but it doesn't answer anything:

    Code (Text):

    In[20]:= Dsolve[2*y + y'^2 + a*x^b == 0, x]

    Out[20]= Dsolve[a x^b + 2 y + (Derivative[1][y])^2 == 0, x]
    Can you help me?
     
  2. jcsd
  3. Feb 27, 2016 #2

    jedishrfu

    Staff: Mentor

    What is this equation for? Is this a home assignment?
     
  4. Feb 28, 2016 #3
    Is not home assignment. Is personal curiosity. I want it to find the radius of a surface of revolution from which I only know the area. After some manipulation, I reduced the equation to this form. ¿Do you want to know how I got it?

    Meanwhile I have been told to look on the Weierstrass function, so I found this paper which shows the solution for a similar case (equation 3.9 page 445, with [itex]-C_0=2[/itex] and [itex]a_2=1[/itex] ), but I can't manage the term [itex]a.x^b[/itex], where [itex]-4 \leq b \leq -2[/itex]. It would be helpful at least find the solution for [itex]b=-3[/itex].
     
    Last edited: Feb 28, 2016
  5. Feb 28, 2016 #4

    jedishrfu

    Staff: Mentor

    Yes, we'd be curious how you came up with the equation.
     
  6. Feb 28, 2016 #5
    Ok. I was reading about Hausdorff dimension, and I wanted to find a continuous, compact surface, without holes, with dimension [itex]1 \leq r \leq 2[/itex].

    So, the area function A(x), to have a Hausdorff dimension r, must satisfy this equation: [itex]\epsilon A_{(\frac{x}{\epsilon})} = \epsilon^r A_{(x)}[/itex].
    Equivalently:

    [itex]A_{(\frac{x}{\epsilon})} = \epsilon^{r-1} A_{(x)} [/itex] [I]

    The derivative of A(x) is
    [itex]A'_{(x)}=\lim_{\epsilon \to 1}\frac{A_{(x.\epsilon)}-A_{(x)}}{x.\epsilon-x} [/itex] [II]

    Because [I]
    [itex]A_{(x.\epsilon)}=\epsilon^{1-r} A_{(x)}[/itex]

    So [II] is
    [itex]A'_{(x)}=A_{(x)} \lim_{\epsilon \to 1}\frac{\epsilon^{1-r}-1}{x.\epsilon-x} = A_{(x)} \frac{1-r}{x}[/itex]

    Solving it: [itex]A_{(x)}=c_1 x^{(1-r)}[/itex]

    I think that is the only solution for [I], but do not have proof.

    I now want this area to continuously transform from the real line (dimension r=1) to the complex plane (r=2), so I want a 2D surface curved in 3D space with area A(x).

    I want to find the function radius R(x) whose surface of revolution is A(x).

    My equation is [itex]A_x =2\pi \int R_{(x)} \sqrt{1+\left( R'_{(x)}\right)^2} \, dx[/itex] [IV]
    I will deal with integration limits later.

    Deriving [IV] and squaring both sides, I get:

    [itex]\left (\frac {c_1 (1-r)}{2 \pi} \right)^2 x^{-2r} = R_{(x)} ^2 + \left ( R_{(x)} R'_{(x)} \right )^2 [/itex]

    Replacing

    [itex]R_{(x)}^2 =2y [/itex]
    [itex] R_{(x)}R'_{(x)} =y' [/itex]
    [itex]a= - \left (\frac {c_1 (1-r)}{2 \pi} \right)^2 [/itex]
    [itex]b= - 2r [/itex]

    I get the differential equation I want to solve:
    [itex]2y + y'^2 + a x^b=0[/itex]

    I had been vaguely told that should look on Weierstrass[/PLAIN] [Broken] p function, [itex]\wp[/itex] and that it may have a closed solution for [itex]b \in \mathbb{Z}[/itex].

    I found this paper, about using the [itex]\wp[/itex] function for solving differential equations, but I can't figure how to use it.
     
    Last edited by a moderator: May 7, 2017
  7. Feb 28, 2016 #6
    Do anybody knows any more specialized forum where I can ask about this?
     
  8. Feb 28, 2016 #7
    Why you asked? Does the answer makes a difference? Does it help to classify the problem or know what area should it be related? Can you tell me at least what information does that gives or could give to you?

    I ask because it may help me to investigate it.
     
  9. Feb 28, 2016 #8

    TeethWhitener

    User Avatar
    Science Advisor
    Gold Member

    Maybe I'm doing something really dumb, but:
    $$2y+(y')^2+ax^b=0$$
    Let ##y=x^2##. Then we have
    $$2(x^2)+(2x)^2+ax^b = 0$$
    Choose ##a=-6,b=2##.
    Maybe that's a start?
     
  10. Feb 28, 2016 #9
    It helps. It says that for dimension ##r=-1: c_1=36\pi## and helps me think out the box of treating b as a variable.

    Actually, those values for a and b have 10 different solutions for y(x). I'm not sure about what to make of it. Maybe one of those solutions is x².

    Thanks you.
     
  11. Feb 28, 2016 #10
    No. It doesn't works. If I choose r=1, then A(x)=constant (which is correct), but R(x) is variable using this method.
     
    Last edited: Feb 28, 2016
  12. Jul 12, 2016 #11
    You can let [tex]w = -y'[/tex] and reduce the equation to an Abel Equation: [tex]ww'-w = f'/2, f = -ax^b[/tex] Unfortunately, known closed form solutions of Abel's equations only exist in special situations, and this isn't one of the known ones.

    Here is a paper on ways to approach Abel's equations: https://arxiv.org/pdf/1503.05929.pdf
     
    Last edited: Jul 12, 2016
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