# Differential Equation arising from Anharmonic Oscillator

1. Oct 29, 2013

### bluesquare

1. The problem statement, all variables and given/known data
Okay, I am trying to solve this Anharmonic Oscillator equation. Now I am given with the potential
$$U=(1/2)x^2-(1/4)x^4$$
and Kinetic energy
$$T=(1/2)x' ^2$$

So the Lagrangian becomes
$\mathcal L$$$=T-U$$

Now I have taken all the $$k$$'s and $$m$$ to be 1
2. Relevant equations
After solving the Lagrangian Equation I got
$$x''(t)=-x(t)+x(t)^3$$

3. The attempt at a solution
And when I used the solution $$x(t)=tanh(t/\sqrt{2})$$ it seems to be satisfying. But my problem is to find a general solution in where I can express my total energy as initial value condition by mentioning my energy and there by controlling how the system behaves e.g how it's position and velocity depends on total enegy sort of like SHO problem where $$x(t)=\sqrt{2E}sin (t)$$ and $$v(t)=\sqrt{2E}cos (t)$$

I want my Anharmonic Oscillator position and velocity to be represented like this where $$E$$ and $$t$$ clearly providing the initial conditions.

Thank you for the time.

2. Oct 29, 2013

### hilbert2

The general solution of the nonlinear DE $\frac{d^{2}x}{dt^{2}}=-x+x^{3}$ is rather complicated and has to be written in terms of elliptic functions. Are you really being asked to find it in your homework?

3. Oct 29, 2013

### bluesquare

Yes this is my homework and I know it involves Elliptic Functions. I've tried it putting that equation In Wolfarm Alpha. It gave me back a huge solution .