= Solution
The <Wronskian> is
$$
W=y_1y_2'-y_1'y_2.
$$
On an interval where $p,q$ are continuous, \b[the solutions are linearly independent exactly when $W(x_0)\ne0$ at one point], equivalently when the <Wronskian> is nonzero throughout the interval. Differentiation and the <homogeneous linear differential equation> give
$$
W'=y_1y_2''-y_1''y_2
=y_1(-py_2'-qy_2)-(-py_1'-qy_1)y_2=-pW.
$$
Hence <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.
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