# Rewriting ODE's into lower orders

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1. Nov 15, 2016

### Euler2718

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

Express

$$\frac{d^{2}x}{dt^{2}} + \sin(x) = 0$$

In a system in terms of $x'$ and $y'$.

2. Relevant equations

3. The attempt at a solution

I seen this example:

$$x^{\prime\prime\prime} = x^{\prime}(t)\cdot x(t) - 2t(x^{\prime\prime}(t))^{2}$$

Where they then wrote:

$$x^{\prime} = y$$
$$y^{\prime} = z$$
$$z^{\prime} = y^{\prime\prime} = x^{\prime\prime\prime} = x^{\prime}(t)\cdot x(t) - 2t(x^{\prime\prime}(t))^{2}$$
So:
$$z^{\prime} = xy - 2tz^{2}$$
Since $y^{\prime\prime} = x^{\prime} = z$

And thus an Euler's method can be devised in MATLAB, given some IVP's of course. Is this then the correct approach for my problem:

$$x^{\prime} = y$$
$$y^{\prime} = x^{\prime\prime} = -\sin(x)$$

Not really familiar with ODE's and such processes, but I need to apply this correctly to proceed with my work.

2. Nov 15, 2016

### Staff: Mentor

This is it right here.

3. Nov 15, 2016

### Euler2718

Thanks. I really had to be sure before continuing.