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!

promitheus said:

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?

Click to expand...

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."

ksdhart2 said:

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."

Click to expand...

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!