Power series solution to a differential equation

[swtlilsoni]

Homework Statement

Using a power series solution, what is the solution to:
(x^2-1)y" + 8xy' + 12y = 0

Homework Equations

Normally these questions specify (about x₀=0) but this one doesn't specify about which point. So if I use the power series equation, what am I supposed to plug in for x₀?

 

[Sethric]

For your x_0, I assume you are talking about an ordinary point about which to expand your series? If so, in this case you can just choose one. I think you will find that x_0 = 0 works. Then just plug in a nice power series. You know the requirements for an ordinary point if you are not given one?

 

[swtlilsoni]

Right given an equation p(x)y" + q(x)y' + r(x)y=0,
an ordinary point is any point at which p(x₀) does not equal 0.

Thus in this equation, the singular points are -1, and 1.
Does that mean I can choose any number that works as an ordinary point and plug it in for x₀? I can choose 9? And plug that in as:
y(x)=\Sigmaa_n(x-9)ⁿ ?

 

[Dick]

Right given an equation p(x)y" + q(x)y' + r(x)y=0,
an ordinary point is any point at which p(x₀) does not equal 0.

Thus in this equation, the singular points are -1, and 1.
Does that mean I can choose any number that works as an ordinary point and plug it in for x₀? I can choose 9? And plug that in as:
y(x)=\Sigmaa_n(x-9)ⁿ ?

Yes, you can chose 9. But why? You have x's in the equation. It's going to be a bit easier just to choose x0=0.

 

[swtlilsoni]

Okay thank you. I did not know I had the freedom to choose any point. I thought the answer varied depending on the chosen point so I thought it needed to be specified.