Where the unknown hides matters
In the previous guide you met the order of a PDE and its principal part — the cluster of highest-order derivative terms that carries the equation's true character. This guide uses exactly that idea to sort PDEs onto a four-rung ladder: linear, semilinear, quasilinear, and fully nonlinear. The single question that climbs the ladder is simple: how does the unknown function u sneak into those highest derivatives?
Picture the equation as a machine. You feed in u and its derivatives, and the machine must output zero. On the lowest rung the machine treats the derivatives gently — it multiplies them only by known functions of the independent variables and adds them up. On higher rungs the machine starts letting u and its lower derivatives reach back in and multiply the higher ones, until on the top rung even the highest derivatives get squared, divided, or stuffed inside a sine. Each step up costs you a structural guarantee.
The bottom two rungs: linear and semilinear
A linear PDE is the friendliest case. Every appearance of u and its derivatives is to the first power and never multiplied by another, and every coefficient depends only on the independent variables — never on u. The heat equation u_t = k u_xx, the wave equation u_tt = c^2 u_xx, and Laplace's equation Laplacian u = 0 are all linear. Because nothing in them bends u against itself, two solutions can be added: this is the superposition principle, the subject of the next guide and the single most powerful tool in the whole linear world.
Climb one rung and you reach the semilinear PDE. Here the principal part — the highest-order terms — is still linear with coefficients that depend only on the independent variables. The nonlinearity is quarantined into the lower-order terms. A clean example is u_t = u_xx + u^3: the top term u_xx is untouched, but the source u^3 is wildly nonlinear. A semilinear PDE keeps the well-behaved 'skeleton' of a linear equation while letting the lower-order 'flesh' misbehave, which is often enough to break superposition yet keep the classification of type intact.
Quasilinear: the unknown bends the principal part
On the third rung the coefficients of the highest derivatives are finally allowed to depend on u and its lower derivatives — but the highest derivatives themselves still appear linearly, to the first power, never multiplied together. This is a quasilinear PDE. The defining feature: u may multiply u_xx, but u_xx never multiplies u_xx. The equation is still 'linear in the leading derivatives,' just with coefficients that move as the solution moves.
The price for that flexibility is steep and concrete. Consider Burgers' equation u_t + u u_x = 0, the prototype quasilinear first-order PDE: the coefficient u multiplies u_x, so the speed at which a value of u travels depends on u itself. Taller parts of the wave move faster, catch up with shorter parts ahead, and the graph steepens until it goes vertical. At that instant a classical, differentiable solution simply ceases to exist — this is shock formation, and it can happen in finite time even from perfectly smooth, gentle initial data.
Fully nonlinear: the top of the ladder
On the top rung the highest derivatives themselves enter nonlinearly — squared, multiplied together, divided, or buried inside another function. A fully nonlinear PDE such as the eikonal equation (u_x)^2 + (u_y)^2 = 1, or the Monge–Ampère equation u_xx u_yy - u_xy^2 = f, can no longer be written as 'linear-in-the-top-derivatives times anything.' Here even the type — elliptic versus hyperbolic — can switch as you move from point to point, depending on the solution you are looking at.
linear a(x) u_xx + b(x) u_x + c(x) u = f(x) semilinear u_xx + g(x, u, u_x) = 0 quasilinear a(x, u, u_x) u_xx + b(x, u, u_x) = 0 nonlinear F(x, u, u_x, u_xx) = 0 (top deriv enters nonlinearly)
Why bother with such a hierarchy? Because it predicts which tools survive. Superposition lives only on the linear rung. The geometric method of characteristics reaches up through the quasilinear rung for first-order equations. For fully nonlinear first-order PDE we need the Hamilton–Jacobi machinery and, beyond classical solutions, the notion of a viscosity solution just to say what 'solving' even means. The higher you climb, the more you trade clean formulas for hard-won existence theorems.
Linearity, honesty, and what the ladder does not promise
Two honest cautions. First, the labels classify the equation as written, not the difficulty of every question about it: a perfectly linear PDE on a nasty domain can be brutally hard, while some nonlinear equations have famous explicit solutions. Second, 'linear' is not the same as 'easy' or 'solvable in closed form' — linear only guarantees that you may add solutions and scale them, which is exactly what the superposition principle cashes in. That guarantee is worth a great deal, but it is one specific gift, not a promise that an answer exists in elementary functions.
There is also a subtler point about homogeneous versus inhomogeneous equations, which sits one axis away from this ladder. Linearity is about how u enters; homogeneity is about whether u = 0 is a solution, i.e. whether there is a forcing term f on the right. A linear inhomogeneous PDE still enjoys a weaker superposition — any two solutions differ by a solution of the homogeneous version — which is precisely the structure the next guide unpacks. Nonlinear equations enjoy no such bookkeeping at all.