Prove for a non-vanishing Wronksian
$y_1(x) = y_{2}(x) \int_{}^{x} \frac{W(\tilde{x})}{y_1(x)^2} \, d\tilde{x}$
I start with normal definition of W, to use find what is $\frac{y_1}{y_2}$:
$W = y_1y_2' - y_1'y_2$
Divide by $y_1^2$:
$\frac{W}{y_1^2} = \frac{y_1y_2'}{y_1^2} - \frac{y_1'y_2}{y_1^2}$
Now if I move the LHS to the RHS, I find:
$ = \frac{y_1y_2'}{y_1^2} - \frac{y_1'y_2}{y_1^2} - \frac{y_1y_2'}{y_1^2} + \frac{y_1'y_2}{y_1^2}$
Which reduces to:
$=\frac{y_2'}{y_1} - \frac{y_2'}{y_1}$
But then that's zero, and they are supposed to be non-zero and linearly independent.
I tried assuming they add rather than subtract, but I am not sure what it leads to later.