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Thread: 2nd order ODE including a quadratic

  1. #1
    Dec 2010

    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?

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  2. #2
    MHF Contributor

    Apr 2005
    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'= $\displaystyle \frac{dy}{dt}= \frac{dy}{dx}\frac{dx}{dt}= y\frac{dy}{dx}$.

    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:
    $\displaystyle y\frac{dy}{dx}= f(x)$
    $\displaystyle \int y dy= \int f(x)dx$

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

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

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

    How difficult those integrals are to do depends strongly on the function f(x).
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  3. #3
    Dec 2010
    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) ?
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