superposition of forcing terms
Suppose a circuit is driven by two sources at once — a steady battery plus an oscillating signal. You could try to handle the messy combined forcing in one go, but linearity offers something easier: solve for each source separately and add the answers. That is superposition of forcing terms, a labour-saving consequence of the equation being linear.
Precisely: if g(x) = g1(x) + g2(x), and y_p1 is a particular solution of y'' + p y' + q y = g1 while y_p2 is a particular solution of y'' + p y' + q y = g2, then y_p1 + y_p2 is a particular solution of the combined equation with forcing g1 + g2. This holds because the operator L[y] = y'' + p y' + q y is linear: L[y_p1 + y_p2] = L[y_p1] + L[y_p2] = g1 + g2. So a complicated right-hand side can be broken into pieces, each tackled with whatever method suits it.
In practice this is how you keep undetermined coefficients tidy when g has several different shapes — say g = 3e^(2x) + 5 sin(x). Instead of one giant trial solution, find y_p for the e^(2x) part and y_p for the sin(x) part separately, then add. It also clarifies modelling: each physical input contributes its own response, and the total response is just the sum. Remember it is the FORCING that superposes here; you still add the single complementary solution y_c only once at the end.
For y'' + y = 4 + 2 sin(x): solve y'' + y = 4 to get y_p1 = 4, and y'' + y = 2 sin(x) (a resonance case needing the x-rule) to get y_p2 = -x cos(x). The combined particular solution is y_p = 4 - x cos(x).
Split a multi-part forcing, solve each piece, add the particular solutions.
Superposition applies to FORCINGS of one and the same linear equation; it does not let you add solutions of two DIFFERENT equations, and it never applies to nonlinear equations.