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| (1) |
Then 0 is a singular point of this equation. Newton polygon is:
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| (2) |
There are slopes > 0 so 0 is an irregular singular point.
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yields two transformed differential equations:
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| (4) |
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| (5) |
These operators have a Newton polygon with slope 0:
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| (6) |
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| (7) |
This can help to find closed-form solutions:
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| (8) |
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| (9) |
Since the general solution of the regular part is a+b*x+c*x^2 for some constants a,b and c, we obtain the general solution of the original equation by taking into account the exponential transformation:
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| (10) |