Suppose the order of a PDE is 2 and the PDE is a linear PDE. In other words, a second-order linear partial differential equation in two variables. Then we can express this PDE as follows:

Au_xx + Bu_xy + Cu_yy + Du_y + Eu_x + Fu = G(x,y)----(eqn 1)
Where A, B, C, D, E, F and G can be constants or functions of x and y

The above equation (eqn 1) is a homogeneous PDE if the right-hand side is equal to zero for all x and y. Otherwise it is a nonhomogeneous PDE.