The question states:
Consider the differential equation $x^2y''+xy'-y=0$. If $y_1$ and $y_2$ are two linearly independent solutions to the differential equation then choose the incorrect:
(1) $W(y_1,y_2)$ is continuous on $\mathbb{R}$
(2) It has unique solution if $y(0)=0,y'(0)=1$
(3) It has no solution if $y(0)=0=y'(0)$
(4) It has unique solution if $y'(1)=0=y(1)$
This is a Cauchy-Euler homogenous equation, so its solution I got as $y=c_1x+\frac{c_2}{x}$.
It is given the option (1) and (3) are incorrect.
The Wronskian $W(y_1,y_2)=Ae^{-\int x/x^2\, dx}=Ae^{-\int 1/x\,dx} =Ae^{-\log x}=\frac{A}{x}$.
And this is not continuous at $0$, so option (1) here is incorrect.
But if we see the obtained solution, we can see $y(x)$ is not even defined at $0$, so how can we use the condition given as $y(0)=0$?
And if options (1) and (3) are the only answer, then options (2) and (4) must be correct. Option (4) can be checked easily and is giving unique solution, but what about (2)? The solution is undefined at zero, how can I say it has unique solution for $y(0)=y'(0)=0$?
Thanks in advance!!