Past exam of the mathematical methods I course of the University of Oxford / 2019 resit / 12 / c / Solution

Linear dependence means there are constants $c_1,c_2$, not both zero, with $c_1{u}_1+c_2{u}_2=0$. Since $u_1$ is nowhere zero, $c_2\ne 0$, so $u_2=C{u}_1$ and $u_2^{\prime }=C{u}_1^{\prime }$. Substitution gives $W=u_{1}C{u}_1^{\prime }-C{u}_1{u}_1^{\prime }=0$.

The nonvanishing assumption is more than is needed for this direction. It also gives the converse directly: if $W=0$, then $(u_2/u_1{)}^{\prime }=W/u_1^2=0$, so the ratio is constant on a connected interval. For arbitrary differentiable functions with zeros, a vanishing Wronskian alone needs more care; here the homogeneous linear equation also supplies an appropriate uniqueness theorem.