
1292 LINEAR PARTIAL DIFFERENTIAL EQUATIONS WITH MAPLE
For linear second-order PDEs, we consider the classification of equations (that does not
depend on their solutions and is determined by the coefficients of the highest derivatives)
and the reduction of a given equation to appropriate canonical forms.
Let us introduce the new variables a=F
p
, b=
1
2
F
q
, c=F
r
, and calculate the discriminant
δ=b
2
−ac at some point. Depending on the sign of the discriminant δ, the type of the
equation at a specific point can be parabolic (if δ=0), hyperbolic (if δ > 0), and elliptic (if
δ < 0). Let us call the equations
u
y
1
y
2
= f
1
(y
1
,y
2
,u,u
y
1
,u
y
2
), u
z
1
z
1
−u
z
2
z
2
= f
2
(z
1
,z
2
,u,u