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Thread: Does this equation makes sense? If so, how to solve it?

  1. #1
    MHF Contributor arbolis's Avatar
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    Does this equation makes sense? If so, how to solve it?

    I'm wondering if the following equation makes sense, and how to solve for $\displaystyle q(t)$.

    $\displaystyle -\frac{dt^2}{LC}=\frac {d^2q}{q}$.

    I found this equation when trying to solve the following problem : http://www.physicshelpforum.com/phys...pacitance.html.
    I wanted to find $\displaystyle V(t)$. I realize the solution should be sinusoidal, so something like $\displaystyle V(t)=V_{max} \sin (\omega t + \phi)$.
    Regarding the problem, when the capacitor halved its energy, I've found out that $\displaystyle V(0.0021s)=\frac{V_{max}}{\sqrt 2}$. So I've found the boundary values and if I find $\displaystyle V(t)$ I'm done.
    If I find $\displaystyle q(t)$, I have that $\displaystyle V(t)=\frac{q(t)}{C}$ and so I'm done.
    I used Kirchhoff's law of voltage to find $\displaystyle -\frac{dt^2}{LC}=\frac {d^2q}{q}$. I might have done errors though.

    Thank you for any help!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by arbolis View Post
    I'm wondering if the following equation makes sense, and how to solve for $\displaystyle q(t)$.

    $\displaystyle -\frac{dt^2}{LC}=\frac {d^2q}{q}$.

    I found this equation when trying to solve the following problem : http://www.physicshelpforum.com/phys...pacitance.html.
    I wanted to find $\displaystyle V(t)$. I realize the solution should be sinusoidal, so something like $\displaystyle V(t)=V_{max} \sin (\omega t + \phi)$.
    Regarding the problem, when the capacitor halved its energy, I've found out that $\displaystyle V(0.0021s)=\frac{V_{max}}{\sqrt 2}$. So I've found the boundary values and if I find $\displaystyle V(t)$ I'm done.
    If I find $\displaystyle q(t)$, I have that $\displaystyle V(t)=\frac{q(t)}{C}$ and so I'm done.
    I used Kirchhoff's law of voltage to find $\displaystyle -\frac{dt^2}{LC}=\frac {d^2q}{q}$. I might have done errors though.

    Thank you for any help!
    It looks like you have tried separating variables for what is essentially a second order homogeneous constant coefficient linear ODE. That is not the way to solve this, you normally use a trial solution of the form $\displaystyle q(t)=e^{\lambda t}$ which will give a quadratic for $\displaystyle \lambda$. Alternatively recognise the ODE as that of SHM and just write down the solution.

    CB
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  3. #3
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    It looks like you have tried separating variables for what is essentially a second order homogeneous constant coefficient linear ODE. That is not the way to solve this, you normally use a trial solution of the form $\displaystyle q(t)=e^{\lambda t}$ which will give a quadratic for $\displaystyle \lambda$. Alternatively recognise the ODE as that of SHM and just write down the solution.

    CB
    Thank you very much!
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