= Solution
For the stated order of the two solutions, define their <Wronskian> by
$$
W=y_1y_2'-y_1'y_2=\det\begin{pmatrix}y_1&y_2\\y_1'&y_2'\end{pmatrix}.
$$
Taking a <derivative> cancels the two mixed products: $W'=y_1y_2''-y_1''y_2$. Substituting the <second-order linear differential equation> for each second <derivative> cancels the $q$ terms and leaves
$$
\boxed{W'+p(x)W=0,\qquad W(x)=W(x_0)\exp\left(-\int_{x_0}^xp(s)\,ds\right).}
$$
This is the <Abel identity>. On an interval with continuous coefficients, <linear independence> means $W(x_0)\ne0$; hence the <Wronskian> never vanishes anywhere on that interval.
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