would it be ok to guess y = sum c_n(x-2)^n ?
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??
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} img.top {vertical-align:15%;} . 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 , .