How to Simplify Equations Using Perturbation Method?

[KFC]

Suppose I have a equation of the form

F(x, k) = G(x, k)

which is unsolvable analytically. We apply the method of pertubtaion (k is small quantity) and let the first order solution approximated as

$$x = x^{(0)} + kx^{(1)}$$

we need to put this back to the equation and find out $$x^{(1)}$$. If after substitution, F(x, k) will only give constant term or term with k and G(x, k) will give some extra terms of order $$k^2$$. For example,

$$F(x, k) = ax^{(0)} + bkx^{(1)}$$
and
$$G(x, k) = cx^{(0)} + dkx^{(1)} + h k^2 (x^{(0)}-x^{(1)})$$

shall I directly drop the high-order term or let the coefficient of the high-order term to zero? For the later one, we will introduce another condition to solve the equation (seems not correct!?)

 

[HallsofIvy]

When you set x= x⁽⁰⁾+ kx⁽¹⁾, you are saying that the perturbation (measured by k) is small enough that higher powers can be ignored. You drop any term with k² or higher.