Power series method and various techniques


by Jen_Jer_888
Tags: power series, power series method, series, series solution
Jen_Jer_888
Jen_Jer_888 is offline
#1
Jul28-11, 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!
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hunt_mat
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#2
Jul29-11, 04:26 AM
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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.
HallsofIvy
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#3
Jul29-11, 06:24 AM
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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^{n-1}[/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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