the superposition principle
Drop two stones into a still pond and the ripples pass right through each other: where two crests meet they add, where a crest meets a trough they cancel, and afterwards each wave continues as if the other had never been there. Linear systems all share this property — effects simply add — and for differential equations it is called the superposition principle.
Precisely: for a homogeneous linear equation, if y1 and y2 are solutions, then every linear combination c1 y1 + c2 y2 is also a solution, for any constants c1 and c2. The reason is mechanical and worth seeing. Write the equation as L[y] = 0, where L[y] means y'' + p y' + q y, the operation of taking those derivatives and combining them. This L is a linear operator: L[c1 y1 + c2 y2] = c1 L[y1] + c2 L[y2], because differentiation distributes over sums and pulls out constants. So if L[y1] = 0 and L[y2] = 0, then L[c1 y1 + c2 y2] = c1·0 + c2·0 = 0. The combination solves the equation automatically.
Superposition is the engine of the entire linear theory. It lets you build the general solution out of a few building-block solutions; it underlies the fact that the solution space is two-dimensional; and in the forced case it lets you break a complicated forcing term into simple pieces, solve each, and add the answers. Be careful, though: superposition holds only for linear equations. For y'' + y^2 = 0 the sum of two solutions is generally not a solution, because the y^2 term does not distribute over a sum.
For y'' + y = 0, sin(x) and cos(x) are both solutions. By superposition so is 3 sin(x) - 7 cos(x): substituting gives (-3 sin x + 7 cos x) + (3 sin x - 7 cos x) = 0. Any amplitude and phase of oscillation is captured this way.
Combining the two basic oscillations sin and cos produces every possible undamped vibration of the system.
For the homogeneous equation you superpose with arbitrary constants. For a nonhomogeneous equation a different version applies: superposing solutions of separate forcing terms gives a solution of the summed forcing — but you may not add an arbitrary multiple of a particular solution and stay a particular solution.