A homework problem I assigned said: The Wronskian of a second order linear ODE is given by: $$ (x-1)e^x. $$ It then asks whether the functions are linearly independent on the whole real line: yes, it is not $0$ for all $x$.
Then it asks whether the functions are solutions on the whole real line to a second order linear ODE: The answers say no because it is $0$ at $x=1$ and Abel's theorem says they must be nonzero for all $x$ to be solutions.
What am I missing? This seems contradictory. Abel's theorem implies that the Wronskian is $0$ for all $x$ or never $0$. How can that function ever be a Wronskian since it is $0$ at $x=1$, but nonzero elsewhere? Also, how can the functions be linearly independent if their Wronskian is $0$ at $x=1$?