Differential Equation -> Behaviour near these singular points

[s3a]

Differential Equation ---> Behaviour near these singular points

Homework Statement

Problem & Questions:
(a) Determine the two singular points x_1 < x_2 of the differential equation
(x^2 – 4) y'' + (2 – x) y' + (x^2 + 4x + 4) y = 0

(b) Which of the following statements correctly describes the behaviour of the differential equation near the singular point x_1?:
A. All non-zero solutions are unbounded near x_1.
B. At least one non-zero solution remains bounded near x_1 and at least one solution is unbounded near x_1.
C. All solutions remain bounded near x_1.

(c) Which of the following statements correctly describes the behaviour of the differential equation near the singular point x_2?:
A. All solutions remain bounded near x_2.
B. At least one non-zero solution remains bounded near x_2 and at least one solution is unbounded near x_2.
C. All non-zero solutions are unbounded near x_2.

Answers:
(a) x_1 = –2 and x_2 = 2
(b) C
(c) B

Homework Equations

Division by the function of x in front of the second order derivative.

The Attempt at a Solution

I understand how to get x_1 and x_2 (by dividing both sides of the differential equation by the function of x in front of the second order
derivative), but could someone please tell me why the multiple-choice parts are C and B, respectively? I don't get the reasoning/logic behind why those are the correct answers.

Any input would be GREATLY appreciated!

 

[Office_Shredder]

For part (b) you forgot to include option C (I assume it's all nonzero solutions remain bounded just from the pattern of the other options)

 

[s3a]

Oops! Sorry, I corrected it now!

 

[s3a]

I just wanted to say that, despite this thread being 15 days old, I am still interested in getting help, if someone is willing to help me out.

 

[vanhees71]

Have look at the Frobenius method, i.e., expansion of solutions around the singular points in terms of generalized power series.

 

[s3a]

Thanks for the answer, vanhees71, but could you please tell me what I would have to do, in order to answer the multiple choice questions, once I obtained the power series solution?

 

[s3a]

Unfortunately, I'm still stuck.

So far, and I'm not sure if I'm on the right track, I'm thinking that I need to use the ratio test to find the radius of convergence for each singular point, and analyze the inequalities obtained.

Is that much correct? If so, what do I do next? If not, could you please, at least, tell me how to get started, in words, and leave the algebra to me?