homogeneous-plus-particular solution structure
Solutions of a linear equation have a beautiful two-part anatomy, and seeing it once explains a lot. The general solution of y' + p(x) y = q(x) is always a homogeneous part plus a particular part: y = y_h + y_p. The first part carries the freedom (the arbitrary constant), and the second part carries the forcing q(x).
Unpack each piece. The homogeneous solution y_h solves the stripped-down equation y' + p(x) y = 0 (forcing turned off); by separation it equals C e^(-integral p dx), and it contains the one free constant C. The particular solution y_p is any single function that satisfies the full equation y' + p(x) y = q(x) — just one specific solution, no free constant. Adding them, y = y_h + y_p, gives every solution: y_h supplies all the wiggle room and y_p anchors the family to the actual right-hand side. You can check this directly — substitute y_h + y_p and watch the y_h terms cancel to zero while the y_p terms reproduce q(x).
This structure is the small-scale rehearsal of one of the central facts of linear differential equations: for any linear equation, general solution equals complementary (homogeneous) solution plus a particular solution. The same pattern reappears for second-order and higher-order linear equations, where y_h becomes a combination of several homogeneous solutions. The honest scope is that this splitting is a feature of LINEARITY only — for a nonlinear equation you cannot add a homogeneous and a particular solution and expect to land on a solution, because superposition fails.
For y' + 2 y = 6 the homogeneous solution is y_h = C e^(-2x), and a particular solution is the constant y_p = 3 (since 0 + 2(3) = 6). So the general solution is y = 3 + C e^(-2x): a fixed offset plus a decaying free part.
y_h carries the arbitrary constant; y_p pins the family to the forcing; their sum is every solution.
There is no single 'correct' y_p — any particular solution works, and two valid choices differ only by a homogeneous solution, which the free constant C absorbs anyway.