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Differential Equation -> Behaviour near these singular points

  1. Jan 10, 2014 #1

    s3a

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    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.

    Answers:
    (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. jcsd
  3. Jan 10, 2014 #2

    Office_Shredder

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    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)
     
  4. Jan 10, 2014 #3

    s3a

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    Oops! Sorry, I corrected it now!
     
  5. Jan 29, 2014 #4

    s3a

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    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.
     
  6. Jan 29, 2014 #5

    vanhees71

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    Have look at the Frobenius method, i.e., expansion of solutions around the singular points in terms of generalized power series.
     
  7. Jan 30, 2014 #6

    s3a

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    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?
     
  8. Feb 12, 2014 #7

    s3a

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    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?
     
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