= Solution
Differentiate the <Wronskian> using the <product rule> and substitute the common <homogeneous linear differential equation>:
$$
\begin{aligned}
W'&=(y_1y_2'-y_1'y_2)'=y_1y_2''-y_1''y_2\\
&=y_1(-py_2'-qy_2)-(-py_1'-qy_1)y_2=-pW.
\end{aligned}
$$
Multiplying this first-order equation by the <integrating factor> $\exp(\int_{x_0}^x p(t)\,dt)$ and integrating gives <Abel's identity>:
$$
\boxed{W(x)=W(x_0)\exp\!\left(-\int_{x_0}^x p(t)\,dt\right).}
$$
The exponential factor is finite and nonzero for every $x$ in the given <interval>, because $p$ is continuous on the compact interval between $x_0$ and $x$. Hence $W(x_0)\ne0$ implies $W(x)\ne0$ everywhere in that interval. Using an integrating factor avoids dividing by $W$, which would assume the conclusion.
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