superposition for linear first-order equations

Superposition is the rule that lets you build solutions of linear equations by adding simpler ones together, and it is the single property that makes linear theory so orderly. In the first-order setting it says, roughly: responses to combined inputs are just combined responses. You can solve a problem in easy pieces and add the pieces up.

More precisely, consider the linear operator L[y] = y' + p(x) y. Superposition has two faces. First, for the homogeneous equation L[y] = 0: if y_1 solves it then so does any constant multiple c y_1, and if y_1 and y_2 both solve it so does y_1 + y_2 — solutions of the homogeneous equation form a tidy one-dimensional family (just multiples of one basic solution). Second, for forcing: if y_a solves L[y] = q_a(x) and y_b solves L[y] = q_b(x), then y_a + y_b solves L[y] = q_a(x) + q_b(x). So you can split a complicated right-hand side into pieces, solve each separately, and sum the particular solutions. This all follows because L respects sums and scalar multiples: L[a u + b v] = a L[u] + b L[v].

Superposition is why the homogeneous-plus-particular structure exists and why methods like undetermined coefficients can attack a sum of forcing terms one term at a time. It is worth stressing the limit plainly: superposition holds because the equation is LINEAR. For a nonlinear equation such as y' = y^2, the sum of two solutions is generally NOT a solution, so none of this machinery applies — which is a large part of why nonlinear equations are so much harder.