A differential equation is linear when the unknown function and its derivatives appear linearly. A second-order example is

$$a(t)y''+b(t)y'+c(t)y=g(t).$$

The associated homogeneous equation sets the forcing term to zero:

$$a(t)y''+b(t)y'+c(t)y=0.$$

If $y_1$ and $y_2$ solve the homogeneous equation, then every linear combination

$$C_1y_1+C_2y_2$$

also solves it. This is the superposition principle.

For a nonhomogeneous equation, the general solution has the form

$$y=y_h+y_p,$$

where $y_h$ is the general homogeneous solution and $y_p$ is any one particular solution.

Linearity is what allows complementary motion and forced response to be studied separately and then added.