I never fully understood the derivation of the method of variation of parameters. Consider the simple case $y'' + p(x)y' + q(x)y = f(x)\,.$
The homogeneous solution is $y_h=c_1y_1+c_2y_2$ and the particular solution that we guess is $y_p =u_1y_1+u_2y_2$ for some function $u_1(t)$ and $u_2(t)$. Next we take derivatives of the particular solution and substitute those back into the original ODE.
The part done next is the part I don't quite see the justification for. We assume that our functions $u_1$ and $u_2$ will satisfy the constraint $u_1'y_1 + u_2'y_2=0\,,$
and this particular constraint will yield a cleaner result when looking at $y'_p$: $u_1'y_1'+u_2'y_2'=f(x)\,.$
At least in the textbook I'm looking at, the justification for assuming that $u_1$ and $u_2$ satisfy $u_1'y_1 + u_2'y_2=0$ is simply omitted. How do we know that the solutions we're after satisfy that constraint? All other sources I look at just say something like, "okay we are going to impose this constraint. Now moving on ..."
Could someone give me a proper explanation as to why we can make this assumption?