linear versus nonlinear equation
Among differential equations there is one giant fault line that separates the equations we can solve cleanly from the ones that resist us: linear versus nonlinear. Linear equations are the tame, well-behaved ones, where effects simply add up — double the cause and you double the effect. Nonlinear equations are where the rich, surprising, and difficult behaviour of the world lives: turbulence, tipping points, chaos.
Precisely, a differential equation is linear if the unknown function and all of its derivatives appear only to the FIRST power, are never multiplied together, and never sit inside another function like sin, exp, or a square. The most general form looks like a_n(x) y^(n) + ... + a_1(x) y' + a_0(x) y = g(x): each y-term is just a coefficient (which may depend on x) times a single derivative of y, added up. The moment you see y^2, y y', sin(y), 1/y, or (y')^2, it is nonlinear. Note the coefficients may be any functions of the INDEPENDENT variable x — that is still linear; nonlinearity is only ever about the unknown y and its derivatives.
Why the obsession with this distinction? Because linear equations obey the superposition principle — solutions can be added and scaled to build new solutions — and that single fact gives us a complete, systematic theory: we can solve essentially every linear constant-coefficient equation. Nonlinear equations have no such universal method; most cannot be solved in closed form at all, and we turn to qualitative pictures and numerical methods instead. Recognizing which side of the line you are on tells you which entire toolbox you may reach for.
y' + x^2 y = sin(x) is linear: y and y' appear only to the first power, and x^2 and sin(x) involve only the independent variable x. But y' = y^2 is nonlinear (y is squared), and y' + sin(y) = 0 is nonlinear (y sits inside sine), even though both look short.
Linearity is only about how the UNKNOWN y appears — the coefficients in x can be as wild as they like.
Coefficients that depend on x do NOT make an equation nonlinear; sin(x) y' is linear, sin(y) y' is not. The test always concerns the unknown function, never the independent variable.