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Date: 5-7-2018
1616
Date: 27-5-2018
828
Date: 27-5-2018
1217
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(1) |
find the first-order solution using a perturbation method. Write
(2) |
and plug back into (1) and group powers to obtain
(3) |
To solve this equation, keep terms only to order and note that, because this equation must hold for all powers of , we can separate it into the two simultaneous differential equations
(4) |
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(5) |
Setting our clock so that , the solution to (4) is then
(6) |
Plugging this solution back into (5) then gives
(7) |
The equation can be solved to give
(8) |
Combining and then gives
(9) |
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(10) |
where the sinusoidal and cosinusoidal terms of order (from the ) have been ignored in comparison with the larger terms from .
As can be seen in the top figure above, this solution approximates only for . As the lower figure shows, the differences from the unperturbed oscillator grow stronger over time for even relatively small values of .
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تفوقت في الاختبار على الجميع.. فاكهة "خارقة" في عالم التغذية
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أمين عام أوبك: النفط الخام والغاز الطبيعي "هبة من الله"
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المجمع العلمي ينظّم ندوة حوارية حول مفهوم العولمة الرقمية في بابل
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