Given a simple harmonic oscillator with a quadratic perturbation ,

(1) 
find the firstorder solution using a perturbation method. Write

(2) 
so

(3) 
Plugging (2) and (3) back into (1) gives

(4) 
Keeping only terms of order and lower and grouping, we obtain

(5) 
Since this equation must hold for all Powers of , we can separate it into the two differential equations

(6) 

(7) 
The solution to (6) is just

(8) 
Setting our clock so that gives

(9) 
Plugging this into (7) then gives

(10) 
The two homogeneous solutions to (10) are
The particular solution to (10) is therefore given by

(13) 
where

(14) 
and the Wronskian is
Plugging everything into (13),
Now let
Then
Plugging and (21) into (2), we obtain the solution

(22) 
© 19969 Eric W. Weisstein
19990526