Rewrite this second-order one-dimensional equation as a first-order two-dimensional system.

Question

I have a problem, where I need to convert a "second-order one-dimensional equation" as a "first-order two-dimensional system".

I have looked online for help on Youtube and other websites, but haven't come close to getting any help with understanding how to convert the equation into the system.

Here's my equation that I need to convert:

$$y''-c(1-y^2)y'+y=0$$ any help on how to rewrite this equation and understanding how to go about this would be most appreciated.

Thank you so much in advance.

Answer 1

Define $$z := y'.$$ So now $$0 = y''-c(1-y^2)y' + y = z' - c(1-y^2)z + y.$$

Now the second-order equation can be represented by the system of two first-order equations in $y$ and $z$, by rearranging the two above equations: \begin{equation}\tag{1} y' = z \end{equation} \begin{equation}\tag{2} z' = c(1-y^2)z - y. \end{equation}