2.1 Linear Equations

The general form of a linear second-order equation in two independent variables x, y is

(2.1)

where a, b, c, d, e, f, g ? C ² ( ?), ? ? R ² and a ²+ b ²+ c ² ? 0 in ?. If we consider the partial differential operator

(2.2)

then the equation (2.1) is written as

while the homogeneous equation corresponding to (2.1) is

(2.3)

The operator L is linear since the condition (1.2) is satisfied for every pair of functions u ₁, u ₂ ? C ² ( ?) and any constants c ₁, c ₂ ? R. From the linearity of the operator it follows that if

are solutions of the homogeneous equation (2.3), then for every choice of constants c ₁ , ..., c _n the function

is also a solution of (2.3). Furthermore, if u _p is a particular solution of Eq. (2.1), then

Thus

is also a solution of Eq. (2.1) for every choice of constants c ₁, ..., c _n.

We shall now consider the simplest case when the coefficients in Eq. (2.1) are real constants. Assume also that the given function g is a real-valued analytic function in ?. Then in some cases we can obtain the general solution of Eq. (2.3), i.e. a relation involving two arbitrary