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## Homework Statement:

- ##\ddot{\xi}(t)=-b\xi (t)+\cos{(\omega t)}(a-c \xi^2(t))##, where ##a, b, c## are constants

## Relevant Equations:

- -

I have to solve the equation above. I haven't heard about an exact method so I tried to apply perturbation theory. I don't know much about it so I would like to ask for some help.

First I put an ##\epsilon## in the coefficient of the non-linear ##\xi^2(t)## term:

##\ddot{\xi}(t)=-b\xi (t)+\cos{(\omega t)}(a-\epsilon c\xi^2(t))##

I calculated to the first order term and it was diverging. I heard that there are some methods that can help to sum a divergent series. Can you suggest any that can work here?

Also, next I tried to put ##\epsilon## in the exponent of the non-linear term, but when I substitute the series into the equation I don't know how to raise the series to the power ##\epsilon##.

So basically ##(\xi_0+\epsilon \xi_1+\epsilon^2 \xi_2+...)^{2 \epsilon}=?##

First I put an ##\epsilon## in the coefficient of the non-linear ##\xi^2(t)## term:

##\ddot{\xi}(t)=-b\xi (t)+\cos{(\omega t)}(a-\epsilon c\xi^2(t))##

I calculated to the first order term and it was diverging. I heard that there are some methods that can help to sum a divergent series. Can you suggest any that can work here?

Also, next I tried to put ##\epsilon## in the exponent of the non-linear term, but when I substitute the series into the equation I don't know how to raise the series to the power ##\epsilon##.

So basically ##(\xi_0+\epsilon \xi_1+\epsilon^2 \xi_2+...)^{2 \epsilon}=?##