Question

Short, multiple-choice problem about singular points of a differential equation

Answer 1

Hello everyone.

I am trying to do the following problem (and another one that is similar), but I don't know how to do parts (ii) and (iii), and I was hoping that someone could tell me the thought process that I must have when answering those parts, since I don't even know how to start.

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

(ii) 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.

(iii) 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:
(i) x_1 = –2 and x_2 = 2
(ii) C
(iii) B
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Any help would be GREATLY appreciated!