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

1. ## 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 $q(t)$.

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

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

Thank you for any help!

2. Originally Posted by arbolis
I'm wondering if the following equation makes sense, and how to solve for $q(t)$.

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

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

CB

3. Originally Posted by CaptainBlack
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 $q(t)=e^{\lambda t}$ which will give a quadratic for $\lambda$. Alternatively recognise the ODE as that of SHM and just write down the solution.

CB
Thank you very much!