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Math Help - A DE I fell over in a physics problem

  1. #1
    MHF Contributor arbolis's Avatar
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    A DE I fell over in a physics problem

    Hello!
    I had to solve a physics problem (related to gases) and I fell over a DE that I don't know how to solve. (I didn't see its type in Chris' tutorial).
    I must find x(t) such that \frac{d^2x}{dt^2}=Gx(t)+K, where G and K are constants.
    I probably made a mistake, because I don't know if we can solve it algebraically and it should describe a harmonic motion.
    I appreciate any help in solving it. Thank you.
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    Quote Originally Posted by arbolis View Post
    Hello!
    I had to solve a physics problem (related to gases) and I fell over a DE that I don't know how to solve. (I didn't see its type in Chris' tutorial).
    I must find x(t) such that \frac{d^2x}{dt^2}=Gx(t)+K, where G and K are constants.
    I probably made a mistake, because I don't know if we can solve it algebraically and it should describe a harmonic motion.
    I appreciate any help in solving it. Thank you.
    it's a simple non-homogeneous linear differential equation with constant coefficients. if G=0, then integrating twice w.r.t. t will give you x(t). if G \neq 0, then a particular solution is x_p=\frac{-K}{G}.

    so you only need to solve x''=Gx, which has characteristic polynomial r^2=G. so now the answer depends on whether G > 0 or G<0. anyway, i think you can take it from here.
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    Quote Originally Posted by arbolis View Post
    Hello!
    I had to solve a physics problem (related to gases) and I fell over a DE that I don't know how to solve. (I didn't see its type in Chris' tutorial).
    I must find x(t) such that \frac{d^2x}{dt^2}=Gx(t)+K, where G and K are constants.
    I probably made a mistake, because I don't know if we can solve it algebraically and it should describe a harmonic motion.
    I appreciate any help in solving it. Thank you.

    we are trying to solve

    x''-Gx=K

    first we would solve the homogenious equation

    x''-Gx=0

    If G>0

    then we get

    m^2-G=0 \iff m \pm \sqrt{G}

    so x_c=c_1e^{t\sqrt{G}}+c_2e^{-t\sqrt{G}}

    and x_p=Ax^2+Bx+C \implies x_p''=2A

    2A+G(Ax^2+Bx+C)=K \implies A=0,B=0,C=\frac{K}{G}

    So we get x(t)=x_c+x_p=c_1e^{t\sqrt{G}}+c_2e^{-t\sqrt{G}}+\frac{K}{G}

    If G<0 you use a the same proceedure.
    Last edited by TheEmptySet; April 20th 2009 at 02:22 PM. Reason: Wow I am way late :/
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    MHF Contributor arbolis's Avatar
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    Thank you both.
    NCA, I don't understand why you wrote
    so you only need to solve
    . Indeed from it I recognize that the solutions are x(t)=A\cos (\omega t +\phi) but I don't understand why K can be considered as 0. Also I don't really know how to derive that the solutions are under the form x(t)=A\cos (\omega t +\phi) but I'm sure I can find the solution in Chris' tutorial.
    By the way K>0. While G can be anything.
    TheEmptySet, I didn't follow you when you wrote " m" and " x_p" and " x_c". Could you explain a bit more? Thanks in advance.
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    assuming the K is a constant, the particulate solution would just be x=A

    correct?
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    Quote Originally Posted by arbolis View Post
    Thank you both.
    NCA, I don't understand why you wrote . Indeed from it I recognize that the solutions are x(t)=A\cos (\omega t +\phi) but I don't understand why K can be considered as 0. Also I don't really know how to derive that the solutions are under the form x(t)=A\cos (\omega t +\phi) but I'm sure I can find the solution in Chris' tutorial.
    By the way K>0. While G can be anything.
    TheEmptySet, I didn't follow you when you wrote " m" and " x_p" and " x_c". Could you explain a bit more? Thanks in advance.
    Try reading the following:

    Second Oder Constant Coefficient Linear Differential Equations | calculuspowerup.com

    Nonhomogeneous Constant Coefficient Linear Differential Equations | calculuspowerup.com

    http://pythagorean.theano.de/fileadm...iffEQnotes.pdf

    http://flash.lakeheadu.ca/~fting/ODE_review.pdf
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    MHF Contributor arbolis's Avatar
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