a linear PDE
The single most important fork in the whole subject is linear versus nonlinear, and a linear PDE is the friendly side. The intuition: in a linear equation the unknown u and its derivatives only ever appear 'plainly' — multiplied by known coefficients and added together — never multiplied by each other, never squared, never fed through a sine or an exponential. If you double all the inputs you double the output; if you add two situations together, their effects add. That gentle, no-surprises behaviour is what makes linear PDEs so much more tractable.
Precisely, a PDE is linear if it can be written as L[u] = f, where L is a linear differential operator — meaning L[a u + b v] = a L[u] + b L[v] for any constants a, b and functions u, v. In practice this means every term is a coefficient (which may depend on the independent variables x, t but not on u) times u or one of its derivatives, to the first power, with no products among them. So u_t = k u_xx is linear; u_t + x u_x = 0 is linear (the coefficient x is fine — it depends only on an independent variable); but u_t + u u_x = 0 is not, because u multiplies its own derivative, and u_t = u^2 is not, because of the square.
Linearity is the goose that lays the golden eggs because it unlocks the superposition principle: any sum of solutions to the homogeneous equation is again a solution. That single fact is the engine behind Fourier series, eigenfunction expansions, transform methods, and Green's functions — you solve the equation for simple building-block inputs and add the pieces to reach any input. Almost every clean, closed-form solution technique in this subject quietly assumes linearity. Nonlinear equations, where superposition fails, are a wilder and largely separate land.
Linear: u_t = k u_xx, u_xx + u_yy = 0, u_t + sin(x) u_x = e^t (variable coefficients and a forcing term are still fine). Nonlinear: u_t + u u_x = 0 (Burgers, a product of u and a derivative), u_xx + u_yy = u^3, (u_x)^2 + (u_y)^2 = 1 (eikonal).
Linearity allows coefficients to depend on x and t, but forbids u or its derivatives from multiplying each other or being squared.
A common slip: a linear PDE may still have a forcing term f and variable coefficients depending on x, t — that does not break linearity. What breaks it is any dependence on u itself inside a coefficient, or any product/power of u and its derivatives. 'Linear' refers to the unknown u, not to the coefficients.