Thread: use the methods of frobenius to solve

1. use the methods of frobenius to solve

use the methods of frobenius to solve:

2xy"+y'+2y =0

i want to know not just the answer but the steps how to answer it (explanation of the answer)

2. Originally Posted by compufatwa
use the methods of frobenius to solve:

2xy"+y'+2y =0

i want to know not just the answer but the steps how to answer it (explanation of the answer)
The charachteristic equation is,
$2r(r-1) + r+0=0$ thus, $r=0,1/2$.
This means to look for solutions of the form:
$y=\sum_{n=0}^{\infty} a_n x^n$ and $y=\sum_{n=0}^{\infty} a_n x^{n+1/2}$
Can you take it from there?

3. ur point of view is not clear to me can u tell me more info about it

4. Originally Posted by compufatwa
ur point of view is not clear to me can u tell me more info about it
Given the equation,
$x^2p(x)y''+xq(x)y'+r(x)y=0$
Where $p(x),q(x),r(x)$ are analytic functions* on some open interval $(-R,R)$.
The method of Frobenius says that we can look for a solution of the form,
$y=\sum_{n=0}^{\infty}a_n x^{n+r}$ for some number $r$.
Now to find this $r$ we solve the equation,
$r(r-1)p(0) + rq(0) + r(0) = 0$.
So I told you what $r$ has to be.

*)"Analytic" on $(-R,R)$ means the function can be expanded as its Taylor series about any point, in this case about 0.

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