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#### issacnewton

##### Member

- Jan 30, 2012

- 61

\( x^2y''-4xy'+6y=0\) on the interval \((-\infty,\infty) \). Show that \( W(y_1,y_2)=0 \) for every real number x.

I could actually show the above by splitting the interval for \( x>0 \) ,\( x=0 \) and \( x<0 \). Now there is theorem about the wronskian. The set of

solutions \(y_1,y_2,\cdots,y_n \) is linearly independent on \( I\) iff \( W(y_1,y_2,\cdots,y_n)\neq 0 \) for every x in the interval. But in this problem,

\( W(y_1,y_2)=0 \) for every real number x. So the reason for this is that , we have \( a_2(x)=x^2 \) , which is zero when x=0 in the given interval. So

one of the conditions for the theorem is not satisfied here. Thats why we get weird behavior here. Is my reasoning correct ?

Thanks