# B Which method to solve this 2nd order DE?

1. Nov 5, 2016

### MalachiK

I have $\frac{d^2x(t)}{dt^2} + B(x(t))^3 = 0$ for a system where I know the initial conditions and where B is a constant that's constructed from the properties of the system. I would like to find $x(t)$.

I've modelled the system in Python and produced some graphs. I know that $x(t)$ is some periodic function.

Could somebody name the method that I should study so that I can get a solution?

2. Nov 5, 2016

### MalachiK

After some further reading I've got this...
Let $s = x'$ so $s' = x''$

$-Bx^3 = x'' = s' \frac{ds}{dt} = \frac{ds}{dx}\frac{dx}{dt} = \frac{ds}{dx} s$

$-\int{Bx^3 dx} = \int{s ds}$

$\frac{-Bx^4}{4} + C = \frac{s^2}{2}$

2C is just a constant that I'll rename C

$\sqrt{C-\frac{Bx^4}{2}} = s = \frac{dx}{dt}$

Since $x(0) = x_0$ and $x'(0) = 0$ we have $C-\frac{Bx_0^4}{2} = 0$ so $C = \frac{Bx_0^4}{2}$

$\frac{dx}{dt} = \sqrt{\frac{B}{2}}\sqrt{x_0^4-x^4}$

$\sqrt{\frac{B}{2}}\int{\frac{1}{\sqrt{x_0^4-x^4}}dx} = \int{dt}$

I don't know if this is correct, but it seems plausible. Now to evaluate the integral.

3. Dec 8, 2016

### joshmccraney

Could you state the initial conditions? I ask because you may be able to perturb the equation, take a naive expansion, and approximating your solution analytically. It would not be exact but the method can be incredibly close (like a truncated Taylor series).

4. Dec 8, 2016