Solution
= Solution
The <Wronskian> is $W=y_1y_2'-y_2y_1'$. Abel's identity gives
$$
W'=-pW.
$$
If $W\ne0$, the two column <vectors> in the displayed representation form a <basis> of $\mathbb R^2$ at each $x$. Differentiating the first component and comparing it with the second gives
$$
u'y_1+v'y_2=0.
$$
Substitution in the differential equation gives
$$
u'y_1'+v'y_2'=f.
$$
Solving this two-by-two system yields
$$
\boxed{u'=-\frac{y_2f}{W},\qquad v'=\frac{y_1f}{W}.}
$$