Question

Hi everyone! Please help me understand why this answer is False. If W(j,k)(x)=0 for all x in I, then j and k and linearly dependent.

I thought if the Wronskian was zero, then the functions ARE DEPENDENT. Please help, very confused. Thanks :o)

Answer 1

I need a very simple explanation if possible :o)

Answer 2

Hi @Kainui ...care to try and answer this question?

Answer 3

Yeah... I read through this like 5 times... I agree with you haha, it seems like it should be true.

When you say "all x in I" what is I? Maybe there's more to this problem that I'm missing, I haven't really thought too deeply about Wronskians in a while since it's usually pretty obvious if you have linearly independent solutions.

Answer 4

I is the interval

Answer 5

I was thinking that even though the wronskian might be zero and showing dependence, that maybe something else could till prevent that from being true sense t states "all x in I" hat maybe there is some condition or value of x that makes it false

Answer 6

Hmmm... is it possible that j and k have 0 coefficients perhaps? So like although these form a fundamental set of solutions \[\Large j= c_1 \cos x \\ \Large k= c_2 \sin x\] if c1 and c2=0 then for all x the wronskian would equal zero...?