So I never fully understood the derivation of the method of variation of parameters.
Consider the simplest case $$y'' + p(x)y' + q(x)y = f(x)$$
And the homogenous solutions is $y_h=c_1y_1+c_2y_2$ and the guess particular solution is $y_p =u_1y_1+u_2y_2$
What is often done is then we take derivatives of the particular solution and substitute back into the original ODE. What is also often done is that we set a constraint $$u_1'y_1 + u_2'y_2=0$$
and this particular constraint will yield $$u_1'y_1'+u_2'y_2'=f(x)$$
And what is often omitted is the explanation for $u_1'y_1 + u_2'y_2=0$. I am going through a book by Nagle and the writer would just throw this out of nowhere and forces me to accept it without fully understanding why we can do this and how do we know the solutions satisfies that particular constraint.
Going through other sources (probably not reliable), I've found that it has something to with the core concept of simple algebra. Like what is known and what is unknown, though it still confounded me...
It was said that it took Lagrange (the creator) a long time to figure out this method. So could someone give me a proper explanation as to why this is true? All the other sources just says "okay we are going to impose this constraint, next moving on..."