Past exam of the differential equations 2 course of the University of Oxford / 2017 / 1 / a / ii / Solution

Differentiate the Wronskian using the product rule and substitute the common homogeneous linear differential equation:Multiplying this first-order equation by the integrating factor $\exp ({\int }_{x_0}^{x}p(t)dt)$ and integrating gives Abel's identity: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 $W(x_0)\ne 0$ implies $W(x)\ne 0$ everywhere in that interval. Using an integrating factor avoids dividing by $W$, which would assume the conclusion.