Generally when we need to solve differential equations, the question can ask us for solve it for the variable to the first order, or maybe the second order. The problem is that i have no idea how the concept of order should be applicable to this equation:
I can understand that, if we want to solve for x til n order, we need to disregard any terms involving $x^{m}, m>n$ in the differential equation, exp:
"Solve $\ddot{x} + x³ + x\dot{x} + x = 0$ to first order in x"
I would disregard x³. $$\ddot{x} + x³ + x\dot{x} + x = 0 \implies \ddot{x} + x\dot{x} + x = 0$$
The problem is, for example, this cross term $x\dot{x}$. Generally i would disregard it too, but just because it remembers me $x$ to the second power $x\dot{x} \approx x*x$. So in the end I would get "$\ddot{x} + x = 0$".
But if the question was solving for the second order in $x$? Should I consider $x\dot{x}$? What about $\dot{x}^2$?
In general, I have no idea what to do, and this certainly leads me to error, considering terms that I shouldn't, and disregarding terms that should be considered.
So, how to interpret "solve for x for the first/second/$n$ order"? And in general, what terms should I disregard?