Theoretical/non-tedious question about power series solution of y'' + y = 0

[s3a]

1. "Homework Statement
Find a recurrence formula for the power series solution around x = 0 for the differential equation given in the previous problem."

The previous problem says:
"Determine whether x = 0 is an ordinary point of the differential equation y'' + y = 0."

Homework Equations

Power series and related stuff.

The Attempt at a Solution

I have the solutions for both of these problems and I also know how to do them both. My question is just:

If x = 0 was not an ordinary point, what would that mean? Would that mean that I cannot assume a power series solution of the form y = [n=0 to inf] Σ[a_n (x - x_0)^n] (where x_0 = 0 in this case) exists or what?

Any input would be greatly appreciated!
Thanks in advance!

 

[LCKurtz]

1. "Homework Statement
Find a recurrence formula for the power series solution around x = 0 for the differential equation given in the previous problem."

The previous problem says:
"Determine whether x = 0 is an ordinary point of the differential equation y'' + y = 0."

Homework Equations

Power series and related stuff.

The Attempt at a Solution

I have the solutions for both of these problems and I also know how to do them both. My question is just:

If x = 0 was not an ordinary point, what would that mean? Would that mean that I cannot assume a power series solution of the form y = [n=0 to inf] Σ[a_n (x - x_0)^n] (where x_0 = 0 in this case) exists or what?

Any input would be greatly appreciated!
Thanks in advance!

If ##x=0## is a regular singular point, your series has to look like$$
\sum_{n=0}^\infty a_nx^{n+r}$$

 

[s3a]

Thanks for your answer. I have a bit more to ask though.

It seems that I can assume there exists a solution y = Σ from n = 0 to inf of a_n * x^n if x = 0 is an ordinary point. I asked what would happen if x = 0 were not an ordinary point and your answer was good for me but now I want to know what if the question asked me to find the power series solution about x = {any one chosen finite real} and if x = {that same one chosen finite real} is an ordinary point, then would that still mean that a solution of the form y = Σ from n = 0 to inf of a_n * x^n exists?

Also does it have to strictly be a real? I'm not looking to go too deep because my course doesn't require it but I just want to know if the answer is a yes or no since I plan to study this further for fun when I am done school.

 

[LCKurtz]

Thanks for your answer. I have a bit more to ask though.

It seems that I can assume there exists a solution y = Σ from n = 0 to inf of a_n * x^n if x = 0 is an ordinary point. I asked what would happen if x = 0 were not an ordinary point and your answer was good for me but now I want to know what if the question asked me to find the power series solution about x = {any one chosen finite real} and if x = {that same one chosen finite real} is an ordinary point, then would that still mean that a solution of the form y = Σ from n = 0 to inf of a_n * x^n exists?

Also does it have to strictly be a real? I'm not looking to go too deep because my course doesn't require it but I just want to know if the answer is a yes or no since I plan to study this further for fun when I am done school.

If ##x=a## is an ordinary point and you want the expansion around that point, you would use a Taylor series centered at ##a##:$$
y=\sum_{n=0}^\infty c_n(x-a)^n$$