So if $(I,x_1)$ and $(I,x_2)$ with $$x_2(t)=x_1(t)\int_{t_0}^t\frac{1}{x_1(s)^2}e^{-\int_{t_0}^sp(r)dr}ds,\quad (t_0,t\in I)$$are solutions to$$x''+p(t)x'+q(t)x=0$$where $p,q$ are continuous and we want to show that $(I,x_1)$ and $(I,x_2)$ are linearly independent we usually find the determine of the Wronskian matrix and show that is not equal to $0$.
However if one wants to show this (in this setup) without Wronskian how would you approach it? Is there an easy solution?
Two nonzero functions are linearly independent if and only if their ratio is non-constant. In particular, if $x_1(t)$ is not identically $0$, and $x_2(t) = x_1(t) \int_{t_0}^t g(s)\; ds$ where $g(s)$ is continuous and not identically $0$ on an interval where $x_1(t)$ is nonzero, then $x_1(t)$ and $x_2(t)$ are linearly independent.