# How to reduce second order nonlinear differential equations into sets of first order differential equations

I have a nonlinear differential equation of the kind:

$f(\ddot{x},\dot{x},x) =0$

I would like to know if there is always a way to write such a differential equation in a form like:

$\dot{y} = g(y)$

that is to put the equation in the form of a set of first order differential equations. What sort of complications arise in the nonlinear case compared to the linear second order ordinary differential equation? Also, what problems may arise when there are terms like $\dot{x} x$ ?

• 'With a nonlinearity of the kind'? I'm not sure what you mean, you can't deduce anything about an arbitrary function $f$ given $f(\ddot{x}, \dot{x}, x) = 0$, except that $f = 0$. – Mattos Dec 2 '16 at 10:45
• @Mattos I have edited my question. I was not much clear, thanks for point this out. – Mirko Aveta Dec 2 '16 at 11:05
• I think you probably mean $\dot{y} = g(x,y)$, not $g(y)$. – Mattos Dec 2 '16 at 11:48
• No, @Mattos. y is just a collected vector I've named y just to remark that it refers to a system of first order nonlinear differential equations. In the linear case, for instance, $y=(x,z)$ where $z=\dot{x}$. – Mirko Aveta Dec 2 '16 at 11:58