 Solution

The Wronskian isOn an interval where $p,q$ are continuous, the solutions are linearly independent exactly when $W(x_0)\ne 0$ at one point, equivalently when the Wronskian is nonzero throughout the interval. Differentiation and the homogeneous linear differential equation giveHence Abel identity reads $W(x)=W(x_0)\exp (-{\int }_{x_0}^{x}p(s)ds)$. If $W(x_0)=0$, the two initial-data vectors are dependent; the uniqueness theorem for ordinary differential equations then makes the corresponding linear combination identically zero. This proves the converse as well as the necessity.