Let's consider a linear operator $L$ such that $$Ly = f$$ subject to some homogeneous boundary conditions on the interval $[a,b]$. The homogeneous solution is $$y_h(x) = c_1y_1(x) + c_2y_2(x)$$ for real constants $c_1, c_2$. Now, according to my notes in class, we assume that the particular solution is $$y_p(x) = u(x)y_1(x) + v(x) y_2(x)$$ Once we define this function, I see that this constraint to $u(x),v(x)$ was imposed, $$u'(x)y_1(x) + v'(x) y_2(x) = 0 $$ This constraint seems to be key to solve the problem, as after simplifying the equation $Ly_p = f$, we can solve for $u(x),v(x)$ by solving the system: $$ u'y_1' + v'y_2' = \frac{f}{p} \\ u'y_1 + v'y_2 = 0$$
My question is, where did the constraint $$u'y_1 + v'y_2 = 0$$ come from? And why is it valid?