power series and differentual equations guidance

I am working on a post exam worksheet and I want to master before my exam tomorrow. I am a little sketchy so if you can please help i really appreciate!! :)

Q. Find an interval around x = 0 for which the initial value problem: (x-2)y'' +3y = x, y(0) = 0, y'(0)=0 = 1 has a unique solution.

I understand that I need to use power series. this is non cauchy-euler equation. My main concern is the power series and the shifts. Because sometimes, I think i read by ferbini's theorem, you can guess: y = sum Cnx^n or y = sum c_n(x-x0)^n

how do you know which guess to make??

Re: power series and differentual equations guidance

would it be ok to guess y = sum c_n(x-2)^n ?

Re: power series and differentual equations guidance

If you do that, you are going to have trouble with the initial conditions which are at x= 0. I would recommend using and writing the equation as xy''- 2y''+ 3y= x. Now, , [tex]y''= \sum n(n-1)a_nx^{n-2}
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. In the first sum let j= n-1 so that n= j+1, n-1= j, and tex]a_n= a_{j+1}" alt=". Putting those into the equation, we get

. In the first sum let j= n-1 so that n= j+1, n-1= j, and tex]a_n= a_{j+1}" /> and the first sum becomes . In the second sum, let j= n- 2 so that n= j+ 2, n- 1= j+ 1, and [tex]a_n a_{j+ 2} and the sum becomes . In the third sum, let j= n so the sum becomes .

That sets the equation as . Now that those all have x to the "j" power, you can combine them (being careful about the j= 0 term) to get recursion relations for the . The initial conditions tell you that , .

Re: power series and differentual equations guidance

Oh! Ok! I get it, thank you so much!!

I am just having trouble seeing when I factor out terms and when to use! do you know of a thread post that has an explanation of that?