second-order linear equation
Most of the motion you can see around you is governed by second-order equations, because acceleration — the second derivative of position — is what forces produce. A swinging pendulum, a bouncing spring, the current in a radio circuit: each obeys a rule that ties the second derivative of some quantity to the quantity itself and its rate of change. A second-order linear equation is the clean, well-behaved version of such a rule.
In standard form it reads y'' + p(x) y' + q(x) y = g(x), where y is the unknown function, p and q are given coefficient functions, and g(x) is a given forcing term. The word 'linear' is the crucial promise: y and its derivatives appear only to the first power, never multiplied together, never inside a sine or a square. So you will never see y^2, or y times y', or (y')^2 — only sums of (a coefficient) times (y, or y', or y''). That single restriction is what makes the whole elegant theory possible. When g(x) = 0 the equation is called homogeneous; when g(x) is not zero it is nonhomogeneous or forced.
Linearity buys you a remarkable structure. The homogeneous solutions form a two-dimensional vector space: find any two genuinely different solutions and every other one is a combination of them. The general nonhomogeneous solution is then one particular solution plus that whole family. Nonlinear second-order equations enjoy none of this — they are solved case by case, often only numerically — so recognising linearity early tells you which powerful tools are on the table.
y'' + 3 y' + 2 y = 0 is second-order, linear, and homogeneous. y'' + 2 y' + y = cos(x) is linear and nonhomogeneous. But y'' + y y' = 0 and y'' + sin(y) = 0 are second-order yet nonlinear — the first multiplies y by y', the second puts y inside a sine.
Linearity is judged by how y and its derivatives appear — not by the coefficients p, q, g, which may be any functions of x.
Linearity is a condition on the unknown y, not on the independent variable x: p(x), q(x), g(x) may be wildly nonlinear functions of x and the equation is still called linear.