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## Main Question or Discussion Point

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

[tex]x = x^{(0)} + kx^{(1)}[/tex]

we need to put this back to the equation and find out [tex]x^{(1)}[/tex]. 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 [tex]k^2[/tex]. For example,

[tex]F(x, k) = ax^{(0)} + bkx^{(1)}[/tex]

and

[tex]G(x, k) = cx^{(0)} + dkx^{(1)} + h k^2 (x^{(0)}-x^{(1)})[/tex]

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!?)

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

[tex]x = x^{(0)} + kx^{(1)}[/tex]

we need to put this back to the equation and find out [tex]x^{(1)}[/tex]. 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 [tex]k^2[/tex]. For example,

[tex]F(x, k) = ax^{(0)} + bkx^{(1)}[/tex]

and

[tex]G(x, k) = cx^{(0)} + dkx^{(1)} + h k^2 (x^{(0)}-x^{(1)})[/tex]

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!?)