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Power series method and various techniques 
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#1
Jul2811, 08:55 PM

P: 4

I know how to do problems like y' + y = 0 where you can replace y' and y with a series in sigma notation, manipulate and compare coefficients.
But how do you solve a differential by power series that does not also include y or a higher order derivative? For example, y' = (x^2) + 2/x + 3. What power series techniques can be employed here? Any help would be appreciated! 


#2
Jul2911, 04:26 AM

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P: 1,583

You not normally use a power series solution for first order differential equations, they're normally for second order and above. In your example you can integrate straight away to find your solution.



#3
Jul2911, 06:24 AM

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Thanks
PF Gold
P: 39,568

If you must use a power series then write
[tex]y= \sum_{n= 0}^\infty a_nx^n[/tex] so that [tex]y'= \sum_{n= 1}^\infty na_nx^{n1}[/tex] Write the right hand side as a power series in x (in your example, [itex]x^2+ 2/x+ 3[/itex], write 2/x as a power series using the generalized binomial theorem) and compare coefficients of the same power. The only difference is that now, you will have a single equation for each "n" rather than a recursion relation. Of course, there will be no equation involving [itex]a_0[/itex] that's your constant of integration. 


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