Solve ODE: Break into System of ODEs

[Dustinsfl]

Is there a way to break this up into a system of ODEs?
$$
L\ddot{\theta} + \dot{x}\dot{\theta} + \ddot{x}\theta = 0
$$

 

[Ackbach]

It's a little unusual that you have a single second-order ODE in two independent variables, but you could just do this:
\begin{align*}
x_{1}&=x \\
x_{2}&= \dot{x} \\
y_{1}&= \theta \\
y_{2}&= \dot{ \theta} \\
0&=L \dot{y}_{2}+x_{2} y_{2}+ \dot{x}_{2} y_{1}.
\end{align*}

There is no $x$ in the original ODE, so you could theoretically integrate that once immediately by essentially leaving out the first equation.

 

[Opalg]

Is there a way to break this up into a system of ODEs?
$$
L\ddot{\theta} + \dot{x}\dot{\theta} + \ddot{x}\theta = 0
$$

Notice that $\dot{x}\dot{\theta} + \ddot{x}\theta = \frac d{dt}(\dot{x}\theta)$, so (assuming that $L$ is a constant) the equation can be written $\frac d{dt}(L\dot{\theta} +\dot{x}\theta) = 0$. You can integrate this once, to get $L\dot{\theta} +\dot{x}\theta = $ const. But you still have the situation of two dependent variables and only one equation, so I don't see how you can go beyond there without further information.

 

[Dustinsfl]

Notice that $\dot{x}\dot{\theta} + \ddot{x}\theta = \frac d{dt}(\dot{x}\theta)$, so (assuming that $L$ is a constant) the equation can be written $\frac d{dt}(L\dot{\theta} +\dot{x}\theta) = 0$. You can integrate this once, to get $L\dot{\theta} +\dot{x}\theta = $ const. But you still have the situation of two dependent variables and only one equation, so I don't see how you can go beyond there without further information.

I know that acceleration is positive and constant and velocity is positive. Does that offer enough information?

 

[blue_raver22]

Yes, it is possible to break this ODE into a system of ODEs. We can rewrite the equation as:
$$
\begin{cases}
\dot{\theta} = \phi \\
\dot{\phi} = -\frac{\dot{x}\phi + \ddot{x}\theta}{L}
\end{cases}
$$
This system of ODEs can then be solved using numerical methods or analytical techniques. Breaking the original ODE into a system of ODEs can often make it easier to solve and can also provide more insight into the behavior of the system.