Here is another problem which I don't have solution as well as idea
Consider the differential equation $$ y''(x) + P(x)y'(x) + Q(x)y(x) = 0 $$ then set of initial conditions for which above differential equation has no solution is :
$$ y(0)=2 \mbox{ and } y'(0)=1 \\ y(1)=0 \mbox{ and } y'(1)=1 \\ y(1)=1 \mbox{ and } y'(1)=0 \\ y(2)=1 \mbox{ and } y'(2)=2 $$
according to me all above satisfy existence theorem but I don't know what to do as 1 option is correct and why ?