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Differential Equation ---> Behaviour near these singular points
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
Division by the function of x in front of the second order derivative.
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!
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!
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