Hi all! I want to perturbatively solve this equation in [tex]\beta[/tex] at second order in [tex]\alpha[/tex](adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{\beta^{2}}{4}-\frac{3}{8}\beta^{4}\alpha+\frac{2}{3}\beta^{6}\alpha^{2}=\frac{1}{4}[/tex]

I rewrite this formula in this way

[tex]\beta^{2}=1+\frac{3}{2}\beta^{4}\alpha-\frac{8}{3}\beta^{6}\alpha^{2}[/tex]

When I try to solve it perturbatively, I obtain

[tex]\beta^{2}=1+\frac{3}{2}1^{2}\alpha-\frac{8}{3}\alpha^{2}\left(1+\frac{3}{2}1^{2}\alpha\right)^{3}[/tex]

The result is

[tex]\beta^{2}=1+\frac{3}{2}\alpha-\frac{8}{3}\alpha^{2}[/tex]

The square root of which gives

[tex]\beta=1+\frac{3}{4}\alpha-\frac{155}{96}\alpha^{2}[/tex]

I know that the correct result is

[tex]\beta=1+\frac{3}{4}\alpha+\frac{61}{96}\alpha^{2}[/tex]

Clearly, there is something wrong in the second order of my calculus.

Can anyone please tell me where I'm mistaking? Thanks a lot.

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# Solve equation perturbatively

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