TITLE: Justifying an Assumption Made While Deriving the Method of Variation of Parameters
QUESTION [20 upvotes]: 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?

REPLY [12 votes]: I quote here an answer I gave to a similar question. 
The notation is that used here. 

This is closely tied to the method of osculating parameters. 
  Suppose we wish to represent, with constant coefficients, some arbitrary function $u(x)$ with two linearly independent functions $u_1(x)$ and $u_2(x)$,
  $$u(x) = A u_1(x) + B u_2(x).$$
  In general this can not be done. 
  The best we can do is match the value of the function and its derivative at some point $x_0$, 
  $$\begin{eqnarray*}
u(x_0) &=& A u_1(x_0) + B u_2(x_0) \\
u'(x_0) &=& A u_1'(x_0) + B u_2'(x_0).
\end{eqnarray*}$$
  The conditions above determine the osculating parameters, the constants $A$ and $B$.
  $A$ and $B$ will be different depending on the point $x_0$.
  In general this fit will be poor at points far from $x_0$.
The method of variation of parameters involves finding the osculating parameters $A$ and $B$ at every point. 
  That is, we let $A$ and $B$ be functions of $x$. 
  The condition that they are the osculating parameters is that they satisfy 
  $$\begin{eqnarray*}
u_G(x) &=& A(x) u_1(x) + B(x) u_2(x) \\
u_G'(x) &=& A(x) u_1'(x) + B(x) u_2'(x),
\end{eqnarray*}$$
  just as above.
  For the second equation to hold it must be the case that 
  $$A'(x)u_1(x) + B'(x)u_2(x) = 0.$$