find the series solution,power seriesby dp182 Tags: around a point, ode, power series, series solution 

#1
Jul311, 10:52 PM

P: 22

1. The problem statement, all variables and given/known data
2xy''x(x1)y'y=0 about x=0 what are the roots of the indicial equation and for the roots find the recurrence relation that defines the the coef a_{n} 2. Relevant equations 2xy''x(x1)y'y=0 about x=0 assuming the solution has the form y=[itex]\Sigma[/itex]a_{n}x^{n+r} y'=[itex]\Sigma[/itex](n+r)a_{n}x^{n+r1} y''=[itex]\Sigma[/itex](n+r)(n+r1)a_{n}x^{n+r2} 3. The attempt at a solution after plugging into the solution I get 2[itex]\Sigma[/itex](n+r)(n+r1)a_{n}x^{n+r1}[itex]\Sigma[/itex](n+r)a_{n}x^{n+r+1}[itex]\Sigma[/itex](n+r)a_{n}x^{n+r1}[itex]\Sigma[/itex]a_{n}x^{n+r} then I attempt to make all the x's the same and and make the sigma's equal so after doing that I get 2[itex]\Sigma[/itex](n+r+1)(n+r)a_{n+1}x^{n+r}[itex]\Sigma[/itex](n+r1)a_{n1}x^{n+r}[itex]\Sigma[/itex](n+r+1)a_{n+1}x^{n+r}[itex]\Sigma[/itex]a_{n}x^{n+r} I know that I need to replace the 0 under the sigma's with a (1) on terms 1,3 but term 2 is whats throwing me off any help would be great 



#2
Jul411, 02:17 AM

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P: 11,521

The third sum should be positive. Explicitly writing in the limits, you have
[tex]2\sum_{n=0}^\infty (n+r)(n+r1)a_n x^{n+r1} \sum_{n=0}^\infty (n+r) a_n x^{n+r+1} +\sum_{n=0}^\infty (n+r)a_n x^{n+r1} \sum_{n=0}^\infty a_n x^{n+r} = 0[/tex] As you noted, only the first and third sums give you a [itex]x^{r1}[/itex] term, and the second sum doesn't give you an [itex]x^r[/itex] term. Separating those terms out, you have \begin{eqnarray*} &&[2r(r1) + r]a_0x^{r1} + \\ &&[2(r+1)r a_1 + (r+1)a_1  a_0]x^r + \\ &&2\sum_{n=1}^\infty (n+r+1)(n+r)a_{n+1}x^{n+r} \sum_{n=1}^\infty (n+r1)a_{n1}x^{n+r} +\sum_{n=1}^\infty (n+r+1)a_{n+1}x^{n+r} \sum_{n=1}^\infty a_n x^{n+r} = 0 \end{eqnarray*} By assumption, a_{0} isn't equal to 0, so you must have 2r(r1)+r=0. That's your indicial equation. 


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