If $x_1$ and $x_2$ are distinct solutions of the differential equation $\ddot{x}+a_1(t)\dot{x}+a_2(t)=0$, show the solutions are linearly dependent.

Question

We are given that $a_1$ and $a_2$ are continuous functions that have a maximum or a minimum at the same $t$, for some $t\in I$. I tried using Abel's Identity with the Wronskian, showing that $$W'(x_1,x_2)=a_2(t)(x_1-x_2)+W(x_1,x_2)$$ Since it is a non-homogeneous equation, I tried to solve first the homogeneous equation $$W'(x_1,x_2)-W(x_1,x_2)=0$$ Then I got $$W(x_1,x_2)=e^{t-t^*+\ln\left({W(x_1(t^*),x_2(t^*)}\right)}$$ That was integrating both sides from $t^*$ to $t$. However, I got stuck here, because I need to show $W(x_1,x_2)=0$, so $x_1,x_2$ are linearly dependent solutions. Plus, I tried to solve the original differential equation, but it's too messy to find the Wronskian of the particular and general solution of this differential equation.