if $y_1$ and $y_2$ are two linear independent solution of $y''+{\sin xy'+e^{-x}y=0}$ , then they can't have the same max and min.
My attempt, suppose they were, then $W(y_{1},y_{2})=y_{1}y_{2}'-y_{1}'y_{2}$. now because it is min max $y_{1}'=y_{2}'=0$ . so $W(y_{1},y_{2})=0$. which cannot be the case for independent 2nd order ODE's solutions.
hmmm something is fishy, we haven't used the ode at all.
2nd Attempt: had they had the same min max point then again $W(y_1,y_2)=0$ but according to Abel's identity, this means $W(y_1,y_2)=e^{cos(x)}$ which cannot be zero, as cos is bounded and e is non zero for any real value.
Both approaches are correct. If the Wronskian of two solutions of a second order linear ODE is zero at some point, then it is zero everywhere. Thus the solutions are linearly dependent.
Usually there is more emphasis on the contraposition, if the Wronskian is non-zero at one point, then it is non-zero everywhere.