**1. The problem statement, all variables and given/known data**

I have a second-order non-linear differential equation that I am trying to solve. So far I have decomposed it into a system of 2 first-order equations, and have (possibly) determined that it cannot be solved analytically. So I am trying to do a nice numerical approximation using GNU Octave (basically compatible with Matlab, so if you can do it with Matlab please can you help too :-D).

Octave needs the equation expressed as first-order DEs -- which I think I've done -- I'm just not sure how to go about doing the actual approximation.

**2. Relevant equations**

Original second-order DE:

[tex]\frac{d^2 \theta}{dt^2} + \frac{k}{m}\cdot\frac{d\theta}{dt} - g\cdot\sin\theta = 0[/tex]

Note that [tex]k, m, g[/tex] are arbitrary constants (yes, [tex]g[/tex] is 9.81!)

**3. The attempt at a solution**

Substitute:

[tex]\frac{d\theta}{dt} = w[/tex]

Hence:

[tex]\frac{d^2 \theta}{dt^2} = w\frac{dw}{d\theta}[/tex]

[tex]w\frac{dw}{d\theta} + (\frac{k}{m}\cdot w) - (g\cdot\sin\theta) = 0[/tex]

Attempt at Octave program to approximate it a bit:

Code:

```
function wdot = f (w, theta)
g = 9.8
k = 1
m = 0.1
wdot = (g*sin(theta) - (k/m)*w)/w
endfunction
theta = linspace(0, 20, 400);
y = lsode ("f", 1, theta);
plot (y, theta);
```

Any help much appreciated, thanks.