Suppose I have a Linear second order Homogeneous, ode $$a_0(x)y''+a_1(x)y'+a_2(x)y=0~, x\in I$$ Now my doubts are
- Can I calculate Wronskian of the two solutions say $y_1$ and $y_2$ using abel's formula, without imposing the condition that $a_0(x)\neq 0\forall x\in I$, $a_0(x),a_1(x),a_2(x)$ are continuous on $I$. If I can proceed with the calculation without the above conditions imposed, can I then deduce the fact that $W(y_1,y_2)>0$ or $W(y_1,y_2)<0$ and $W(y_1,y_2)\equiv 0$.
Can I also deduce that $W(y_1,y_2)(x_0)\neq 0, $ for some $x_0\in I\implies $ $y_1,y_2$ are L.I. $W(y_1,y_2)(x_0)= 0, $ for some $x_0\in I\implies $ $y_1,y_2$ are L.D.