linear versus nonlinear ODE

This is the single most consequential fork in the road for differential equations — more decisive than order or degree. On one side sit linear equations, the gentle, well-mapped country where almost every clean technique lives. On the other sit nonlinear equations, a wilder territory where solutions can blow up, split, or never be writable in closed form. Knowing which side you are on tells you immediately what tools you may use.

An ODE is linear if the unknown function and all its derivatives appear only to the first power, are never multiplied by each other, and never sit inside a nonlinear function. For first order this means it can be written in the standard form dy/dx + P(x)*y = Q(x): y appears alone, to the first power, with coefficients that depend on x but never on y. Everything else is nonlinear. So y' + x*y = sin(x) is linear; y' = y^2 is nonlinear (y squared); y' = y*y' would be nonlinear (product of unknown quantities); y' = sin(y) is nonlinear (y inside a sine). The coefficients may be ferocious functions of x — that does not break linearity; linearity is a statement about how y and its derivatives enter, not about x.

The reason this matters so deeply is the superposition principle: for a linear equation, sums and scalar multiples of solutions behave predictably, which is exactly what lets us build general solutions, Fourier methods, transforms, and the whole machinery of mathematical physics. Nonlinear equations forfeit that gift; most have no elementary solution and must be attacked qualitatively (direction fields, phase line) or numerically (Euler's method and its successors). The cleverest moves in this field — the Bernoulli and Riccati substitutions — are precisely tricks that turn a special nonlinear equation back into a linear one.