the general-solution structure theorem
Here is the one idea that organizes this whole subject. To solve a forced linear equation you do not need a single clever formula — you need two separate, more manageable pieces, and you simply add them. That is what the structure theorem promises.
Precisely: the general solution of y'' + p(x) y' + q(x) y = g(x) is y = y_c + y_p, where y_c = c1 y1 + c2 y2 is the complementary solution (the general solution of the homogeneous equation) and y_p is ANY one particular solution of the forced equation. Why does this work? Linearity. If y_p and another solution Y both satisfy the forced equation, their difference Y - y_p satisfies the homogeneous equation, so Y - y_p must be some c1 y1 + c2 y2; rearranging gives Y = y_c + y_p. So every solution is captured, and the two arbitrary constants in y_c give you exactly the freedom you need to meet initial conditions.
This is why the two big particular-solution methods — undetermined coefficients and variation of parameters — are worth learning: they each manufacture one y_p, and the theorem assembles the rest. It also explains the physical split: y_c is the transient, y_p the steady state. The theorem holds only because the equation is linear; for nonlinear equations you cannot add solutions this way.
For y'' + y = 2x, one particular solution is y_p = 2x, and the homogeneous part gives y_c = c1 cos(x) + c2 sin(x). The general solution is the sum y = c1 cos(x) + c2 sin(x) + 2x.
General solution = complementary solution + any one particular solution.
Any particular solution will do — two people can write different-looking y_p and still get the same family, because the difference of two particular solutions is just absorbed into y_c.