Differential Equation -> Behaviour near these singular points

1. Jan 10, 2014

s3a

Differential Equation ---> Behaviour near these singular points

1. The problem statement, all variables and given/known data
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.

(a) x_1 = –2 and x_2 = 2
(b) C
(c) B

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

3. 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!

Last edited: Jan 10, 2014
2. Jan 10, 2014

Office_Shredder

Staff Emeritus
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)

3. Jan 10, 2014

s3a

Oops! Sorry, I corrected it now!

4. Jan 29, 2014

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.

5. Jan 29, 2014

vanhees71

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

6. Jan 30, 2014

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?

7. Feb 12, 2014

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?