Solve Diff. Eq. using power series

[JamesonS]

Homework Statement

\begin{equation}
(1-x)y^{"}+y = 0
\end{equation}

I am here but do not understand how to combine the two summations:
Mod note: Fixed LaTeX in following equation.
$$(1-x)\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n+\sum_{n=0}^{\infty}a_nx^n = 0$$

 

[Ray Vickson]

Homework Statement

\begin{equation}
(1-x)y^{"}+y = 0
\end{equation}

I am here but do not understand how to combine the two summations:

$$(1-x)\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n+\sum_{n=0}^{\infty}a_nx^n = 0$$

Here is your equation using PF-compatible TeX:
$$
(1-x)\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2} x^n+\sum_{n=0}^{\infty}a_nx^n = 0
$$
Just replace the "\ begin {equation} ... \ end {equation} " by "$ $ ... $ $ " (no spaces between the initial and final $ signs). Also: write "\infty", not "\infinity".

As for your question: write out the first 3 or 4 terms, to see what you get. That will give you insight into what you should do next.

 

[JamesonS]

Thanks for the response and Latex help. Writing out the first few terms of each sum:
$$
(1-x)[2a_2+6a_3x+12a_4x^2+...]+[a_0+a_1x+a_2x^2+...]
$$
I am not sure what to do with the (1-x) term outside the first sum...

 

[Ray Vickson]

Thanks for the response and Latex help. Writing out the first few terms of each sum:
$$
(1-x)[2a_2+6a_3x+12a_4x^2+...]+[a_0+a_1x+a_2x^2+...]
$$
I am not sure what to do with the (1-x) term outside the first sum...

What is preventing you from "distributing out" the product? That is, ##(1-x) P(x) = P(x) - x P(x).##

 

[MidgetDwarf]

Been a while since I did DE, but doesn't the OP have to be aware of the singular point in this problem? Hence he has to use the method of Frobenius?

 

[LCKurtz]

Been a while since I did DE, but doesn't the OP have to be aware of the singular point in this problem? Hence he has to use the method of Frobenius?

But the singular point isn't at ##x=0##.

 

[MidgetDwarf]

Thanks for the correction. It's been a while. I remembered that there is no singular point if we take the Taylor expansion about x=0? Correct?

 

[epenguin]

I am not sure what to do with the (1-x) term outside the first sum...

Practically after your first equation, or at any later stage, just multiply it out. You have shown that you know how to write a sum of different powers of x in terms of xⁿ by changing the subscript appropriately.

 

[vela]

Thanks for the correction. It's been a while. I remembered that there is no singular point if we take the Taylor expansion about x=0? Correct?

That's backwards. If you expand about a regular point, then the solution can be written as a Taylor series. If there's a singular point, then you can use the method of Frobenius and end up with a Laurent series.

 

[MidgetDwarf]

That's backwards. If you expand about a regular point, then the solution can be written as a Taylor series. If there's a singular point, then you can use the method of Frobenius and end up with a Laurent series.

Thanks Vela!

It has been a while. I may pop open a differentials to bring the memory back.