1. The problem is to write down the general linear partial differential equation (PDE) of second order. 2. A second-order linear PDE involves partial derivatives of a function $u(x,y)$ up to the second order. 3. The general form of a linear second-order PDE in two variables $x$ and $y$ is: $$A(x,y)\frac{\partial^2 u}{\partial x^2} + 2B(x,y)\frac{\partial^2 u}{\partial x \partial y} + C(x,y)\frac{\partial^2 u}{\partial y^2} + D(x,y)\frac{\partial u}{\partial x} + E(x,y)\frac{\partial u}{\partial y} + F(x,y)u = G(x,y)$$ 4. Here, $A$, $B$, $C$, $D$, $E$, $F$, and $G$ are given functions of the independent variables $x$ and $y$. 5. Important notes: - The equation is linear because $u$ and its derivatives appear to the first power and are not multiplied together. - The coefficients can vary with $x$ and $y$. - The highest order derivatives are second order. 6. This form is used to classify PDEs into elliptic, parabolic, or hyperbolic types based on the discriminant $B^2 - AC$. This is the general linear second-order PDE.