Results 1 to 7 of 7

Math Help - strange differntial equation

  1. #1
    Newbie
    Joined
    Feb 2010
    From
    Bucuresti
    Posts
    8

    strange differntial equation

    Hi all,

    can someone tell me what the general solution is for the following differential equation.

    z''+z=b*z^3 (z^3 meaning "z" to the third power, not the third derivative of z, while z'' stands for the second derivative of z)

    Is it possible that there is no solution for it.

    How do you in heavens name solve such differential equations?

    Thanks in advance
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Dec 2009
    From
    1111
    Posts
    872
    Thanks
    3
    Dear polo,

    This differential equation could be solved in several ways. But I will give you the most easiest I can think of. In a differential equation if x is absent explicitily, (does not have x^{1} terms.) this method can be used.

    \frac{d^2z}{dx^2}+z=bz^{3}

    Substitute, p=\frac{dz}{dx}

    \frac{dp}{dx}=\frac{dp}{dz}\times\frac{dz}{dx}=p\f  rac{dp}{dz}

    \frac{dp}{dx}+z=bz^{3}

    p\frac{dp}{dz}+z=bz^{3}

    Seperation of variables and integrating yields, \frac{p^2}{2}=\int{(bz^{3}-z)dz}

    \frac{p^2}{2}=\frac{bz^{4}}{4}-\frac{z^{2}}{2}+C~;Where~C~is~an~arbitary~constant  .

    Hope this will help you.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Feb 2010
    From
    Bucuresti
    Posts
    8
    Quote Originally Posted by Sudharaka View Post
    Dear polo,

    This differential equation could be solved in several ways. But I will give you the most easiest I can think of. In a differential equation if x is absent explicitily, (does not have x^{1} terms.) this method can be used.

    \frac{d^2z}{dx^2}+z=bz^{3}

    Substitute, p=\frac{dz}{dx}

    \frac{dp}{dx}=\frac{dp}{dz}\times\frac{dz}{dx}=p\f  rac{dp}{dz}

    \frac{dp}{dx}+z=bz^{3}

    p\frac{dp}{dz}+z=bz^{3}

    Seperation of variables and integrating yields, \frac{p^2}{2}=\int{(bz^{3}-z)dz}

    \frac{p^2}{2}=\frac{bz^{4}}{4}-\frac{z^{2}}{2}+C~;Where~C~is~an~arbitary~constant  .

    Hope this will help you.
    Thanks Sudharaka,

    I seems to me that there is no particular solution to this differential equation; intuitively it looks to me that we are dealing with a helix since the differential equation z''+z=b is an ellipse and obeys Keplers first law
    My mathematical skills need some revision (it has been a long time)

    Thanks in advance
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Feb 2010
    From
    Bucuresti
    Posts
    8
    Quote Originally Posted by polo View Post
    Thanks Sudharaka,

    I seems to me that there is no particular solution to this differential equation; intuitively it looks to me that we are dealing with a helix since the differential equation z''+z=b is an ellipse and obeys Keplers first law
    My mathematical skills need some revision (it has been a long time)

    Thanks in advance
    Sorry a spiral i mean...it looks like a spiral
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Feb 2010
    From
    Bucuresti
    Posts
    8
    Quote Originally Posted by polo View Post
    Sorry a spiral i mean...it looks like a spiral
    Hi Sudharaka,

    Maybe I should be little bit more clear on my question
    I have include a word document where I have outlined the problem a little bit better.


    I have done it in word because for the moment I don't know how to write mathematical formulas in this text box.


    Thanks in Advance
    Attached Files Attached Files
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member
    Joined
    Dec 2009
    From
    1111
    Posts
    872
    Thanks
    3
    Dear Polo,

    In the above solution I have used you will have you substitute for p and integrate it another time to get the final answer, which is out of my scope. But if you use the Wolfram integrator you could find the solution as,

    http://integrals.wolfram.com/index.j...D&random=false or http://integrals.wolfram.com/index.j...D&random=false

    The answers contains imaginary numbers. So I do not think it will be a circle or an ellipse.

    Hope this will help you.
    Last edited by Sudharaka; February 5th 2010 at 06:04 PM.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Feb 2010
    From
    Bucuresti
    Posts
    8
    Well what should I say...."Thanks".

    Things are becoming clear...any while what a strange function.

    Thanks Sudharaka, thanks again.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Homogenous Differntial Equation
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: February 14th 2011, 10:33 AM
  2. with this differntial equation problem
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: November 9th 2009, 04:08 PM
  3. Help With differntial equation
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: April 30th 2009, 02:27 PM
  4. Differntial Equation Trouble With Integral
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: February 2nd 2009, 08:14 PM
  5. Differntial Equation Help
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 9th 2008, 03:22 AM

Search Tags


/mathhelpforum @mathhelpforum