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Thread: Fundamental set of solutions for a DE: functions cos(ln x) and sin(ln x)

  1. #1

    Arrow Fundamental set of solutions for a DE: functions cos(ln x) and sin(ln x)

    Hi,

    Need some help with an assignment question. I think I have it worked out but I am a little rusty with some basic principles
    Any help will be greatly appreciated.

    Q: Verify that the functions cos(ln x), sin(ln x), form a fundamental set of solutions of the DE x2y′′ + xy′ + y = 0, on the interval (0, ∞). Form the general solution.

    My attempt and my confusion:

    I have verified that both y1(x)= cos(lnx) and y2(x)= sin(lnx) are solutions of the DE.

    I have then tried to verify if W(y1,y2) does not equal 0

    w(y1,y2)= cos(lnx)*(cos(lnx)/x) - ((-sin(lnx)/x)*sin(lnx))
    = (cos^2(lnx) + sin^2(lnx))/x
    = 1/x which does not equal 0

    Confusion: The interval is (0, ∞) and x cannot be 0, so does this mean the Wronskian is discontinuous at x? And would it make the equation linearly dependent?

    Thanks!
    Last edited by Subhotosh Khan; 08-27-2017 at 08:12 PM.

  2. #2
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    Quote Originally Posted by promitheus View Post

    Confusion: The interval is (0, ∞) and x cannot be 0, so does this mean the Wronskian is discontinuous at x? And would it make the equation linearly dependent?
    Erm... sorry, but I don't think I understand what you're asking. You've determined that the Wronskian is 1/x. This function is therefore continuous everywhere except x = 0, which you know x cannot be because of the specified interval. Further, I'm not quite sure what you even mean by "discontinuous at x."

  3. #3
    Quote Originally Posted by ksdhart2 View Post
    Erm... sorry, but I don't think I understand what you're asking. You've determined that the Wronskian is 1/x. This function is therefore continuous everywhere except x = 0, which you know x cannot be because of the specified interval. Further, I'm not quite sure what you even mean by "discontinuous at x."
    Sorry, I meant discontinuous at 0. And I have just realised I mixed up my open and closed intervals. I think I know how to proceed from here.

    Thanks for your help!

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