Results 1 to 3 of 3

Math Help - 2nd order ODE including a quadratic

  1. #1
    Newbie
    Joined
    Dec 2010
    Posts
    2

    2nd order ODE including a quadratic

    Hi all,
    I am trying to solve for the rocking frequency of a semi-circular object (imagine a cylinder cut down the z axis so that the cross section is semi-circular), and I have got stuck right at the end.

    I figured out the location of the centre of mass so that I could write down the torque experienced by the object when sitting at any angle, and am now faced by a differential equation that looks substantially trickier than I imagined.

    Note that I converted sines and cosines to the following:
    sin(x) ---> x
    cos(x) ---> 1 - 0.5x^2

    After all this work I am left with an equation of the form: x'' = f(x)
    Where f(x) contains a linear and quadratic term.

    Can anyone give me any clues as to how I might attack this equation?

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,973
    Thanks
    1121
    Since the independent variable (which I will call "t") does not appear in that equation there is a standard method of reducing the order of the equation (often called "quadrature").

    Let y= x'. Then x''= y'= \frac{dy}{dt}= \frac{dy}{dx}\frac{dx}{dt}= y\frac{dy}{dx}.

    So the second order equation for x as a function of t converts to a first order equation for y as a function of x:
    y\frac{dy}{dx}= f(x)
    \int y dy= \int f(x)dx

    \frac{1}{2}y^2= \int f(x)dx+ C
    (That y^2 is the reason for the name "quadrature".)

    y= \frac{dx}{dt}= \pm\sqrt{2\left(\int f(x)dx+ C\right)}

    \int \frac{dx}{\sqrt{2\left(\int f(x)dx+ C\right)}}= t+ D

    How difficult those integrals are to do depends strongly on the function f(x).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Dec 2010
    Posts
    2
    Thanks HallsofIvy. That's a nice trick I will have to remember.

    Since f(x) is polynomial in x (with only linear and quadratic terms), the first integral should be quite simple to do. The second looks pretty horrible though How to do an integral of the inverse of a third order polynomial?! Sounds like another variable substitution might help. Perhaps I should make life easy for myself, and remove the quadratic term by approximating cos(x)--->1.....

    Is there a name for differential equations of the form: x''=f(x) ?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. sloving for two variables including fractions
    Posted in the Algebra Forum
    Replies: 2
    Last Post: August 18th 2011, 02:09 AM
  2. cdf for funcion including E(X) and V(X)
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: June 27th 2010, 08:21 AM
  3. Proof Shuttle Sort is a quadratic order algorithm.
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: March 11th 2010, 12:50 PM
  4. Word problem including depth, min / max..
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: April 15th 2009, 10:16 AM
  5. indefinite integral, including e
    Posted in the Calculus Forum
    Replies: 8
    Last Post: April 14th 2009, 06:47 AM

Search Tags


/mathhelpforum @mathhelpforum