Find two linearly independent power series solutions

[pinky]

Hi,

My previous post didn't work so I've attached two files for the problem.
I don't really know how to get C2 and the recurrence. I don't fully understand it and have been following examples that I found in my book.
As for the last question, the book says to use c0=0 and C1=1. I answered the "answer is not shown" and both are 0, which are both incorrect.

If someone could help me with this problem that would be great.

Thank you in advance.

 

[Ishuda]

Hi,
My question is:

Find two linearly independent power series solutions of the given differential equation about the ordinary point x = 0.
2y'' + x²y' + xy = 0


Use: y(x) =

∞	cnxn
n = 0

, and use the lower case letter c, for your subscripts in using c.

2y'' + x²y' + xy = 0

...

You work didn't come through. It looks like it is at least partially correct but you will have to find a different way to show us your work. I got c₀ and c₁ are arbitrary and c₂=0 and a solution for the remainder of the coefficients of the form
\(\displaystyle c_{3(n+1)+j}\, =\, \alpha_{3n+j}\, c_{3n+j}\); j =0, 1, 2; n=0, 1, 2,...
where the \(\displaystyle \alpha_{3n+j}\) is determined by the differential equation. However, I haven't checked that answer.

Note that this says
(1) c_j, j=3, 6, 9, ... depends only on c₀
(2) c_j, j=4, 7, 10, ... depends only on c₁
(3) c_j, j=5, 8, 11, ... depends only on c₂ (and thus are 0)

 

[pinky]

You work didn't come through. It looks like it is at least partially correct but you will have to find a different way to show us your work. I got c₀ and c₁ are arbitrary and c₂=0 and a solution for the remainder of the coefficients of the form
\(\displaystyle c_{3(n+1)+j}\, =\, \alpha_{3n+j}\, c_{3n+j}\); j =0, 1, 2; n=0, 1, 2,...
where the \(\displaystyle \alpha_{3n+j}\) is determined by the differential equation. However, I haven't checked that answer.

Note that this says
(1) c_j, j=3, 6, 9, ... depends only on c₀
(2) c_j, j=4, 7, 10, ... depends only on c₁
(3) c_j, j=5, 8, 11, ... depends only on c₂ (and thus are 0)

Hi, thanks for your response and I have fixed the post.

 

[Ishuda]

Hi, thanks for your response and I have fixed the post.

The type is still too small for me, even if I enlarge the image I can't determine what is there.