# Application of another potential to harmonic oscillator

I know how to solve the harmonic oscillator equation,

now, I would like to apply another potential (electric potential) to this equation

therefore,

I have a 2nd order differential equation

$m\frac{d^2x }{dt^2}+b\frac{dx }{dt}+k\frac{1}{x^2}=0$

b is friction constant k is constant for electric potential

is there any way to solve this analytically ?

I tried to get the answer from the derivative table, but can not found the answer.

just giving me an derivative table will be also nice for me.

Thanks

I hope it does have a analytic solution.

just in case, i can not get a solution,

only at x = near zero,

I can assume the force will be zero $(m\frac{d^2x }{dt^2})=0$

so the $\frac{dx }{dt}=-\frac{k}{bx^2}$

is this reasonable ?

• Just $\frac{d^2 x}{dt^2} + \frac{1}{x^2} = 0$ has some hope of solution; multiplying through by $\frac{dx}{dt}$ and integrating gives $\frac{1}{2} x'^2 - 1/x = 0$, which you can exactly integrate. With the friction in there you have a problem, though.
– Ian
Commented Apr 10, 2016 at 3:50
• Thanks, that will be the case when the friction is zero. I am thinking my system will have a lot of friction, so the solution have some behavior like overdamped harmonic oscillator......
– eric
Commented Apr 10, 2016 at 3:57
• Sorry, I made an error, the other side should be $C$, not $0$. Still, for a given choice of a sign of the square root, you can solve $x'=\sqrt{2/x+C}$, it is a separable equation. But yes, with the friction you should expect some difficulty.
– Ian
Commented Apr 10, 2016 at 4:20