(adsbygoogle = window.adsbygoogle || []).push({}); ODE Series Solution Near Regular Singular Point, x^2*y term? (fixed post body)

1. The problem statement, all variables and given/known data

Find the series solution (x > 0) corresponding to the larger root of the indicial equation.

[tex]5x^{2}y'' + 4xy' + 10x^{2}y = 0[/tex]

2. Relevant equations

Solution form:

[tex]y = \sum_{i=0}^{\infty}a_{n}x^{r+n}[/tex]

3. The attempt at a solution

[tex]\lim_{x\to\0}\frac{4x}{5x^{2}}x = \frac{4}{5} < \infty[/tex]

[tex]\lim_{x\to\0}\frac{10x^{2}}{5x^{2}}x^{2} = 2x^{2} = 0 < \infty[/tex]

The singular point is regular, and the solution should be in the form of (and its derivatives):

[tex]y = \sum_{n=0}^{\infty}a_{n}x^{r+n}[/tex]

[tex]y' = \sum_{n=0}^{\infty}a_{n}(r+n)x^{r+n-1}[/tex]

[tex]y'' = \sum_{n=0}^{\infty}a_{n}(r+n)(r+n-1)x^{r+n-2}[/tex]

Plug the solution form into the ODE:

[tex]5x^{2}\sum_{n=0}^{\infty}a_{n}(r+n)(r+n-1)x^{r+n-2} + 4x\sum_{n=0}^{\infty}a_{n}(r+n)x^{r+n-1} + 10x^{2}\sum_{n=0}^{\infty}a_{n}x^{r+n}[/tex]

Multiply the x terms into the series:

[tex]5\sum_{n=0}^{\infty}a_{n}(r+n)(r+n-1)x^{r+n} + 4\sum_{n=0}^{\infty}a_{n}(r+n)x^{r+n} + 10\sum_{n=0}^{\infty}a_{n}x^{r+n+2}[/tex]

Here is where my trouble lies. My understanding is that I am trying to pull out the [tex]a_{0}[/tex] term of the series which will leave behind the indicial equation I am also to combine all of the series into one, factor out the x term, and then find the recurrence relation for the series term.

My problem specifically is the last term, which now has a [tex]x^{r+n+2}[/tex] that I can't fit into the rest of the series.

These are my attempts:

1) Blindly pull out [tex]a_{0}[/tex]:

[tex]5a_{0}(r)(r-1)x^{r} + 4a_{0}r^{r} + 10a_{0}x^{r+2} + 5\sum_{n=1}^{\infty}a_{n}(r+n)(r+n-1)x^{r+n} + 4\sum_{n=1}^{\infty}a_{n}(r+n)x^{r+n} + 10\sum_{n=1}^{\infty}a_{n}x^{r+n+1}[/tex]

Stuck because of the [tex]x^{r+2}[/tex] term in the indicial equation and the last series term [tex]x^{r+n+2}[/tex] is still an issue...

2) Try to combine series first

Can't find a way to shift the indexes so that they are all the same and have an x exponent that will allow me to factor them out after I combine the series.

3) Desperately ignore term 3 and pull out [tex]a_{0}[/tex] from just the first two terms, even though this seems very wrong.

[tex]5a_{0}(r)(r-1)x^{r} + 4a_{0}r^{r} + 5\sum_{n=1}^{\infty}a_{n}(r+n)(r+n-1)x^{r+n} + 4\sum_{n=1}^{\infty}a_{n}(r+n)x^{r+n} + 10\sum_{n=1}^{\infty}a_{n}x^{r+n+1}[/tex]

leaving the indicial equation of:

[tex]a_{0}(5r^{2}-r) = 0[/tex]

[tex] r = 0, \frac{1}{5} [/tex]

This is somewhat rewarding, 1/5 is available choice for the r value (multiple choice question), and r = 0 seems to be correct for the version of the question which ask to find the solution for the lowest r.

But I am still not sure how to combine the series...

I have looked and many examples online but I am unable to find an example which ends up with [tex]r+n+2[/tex] exponent for x and I am thoroughly out of ideas.

Any help would be greatly appreciated.

Thanks!

- Andrew Balmos

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# Homework Help: ODE Series Solution Near Regular Singular Point, x^2*y term?

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