================================================== ==================================================

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

================================================== ==================================================