If z is a function of two independent variables x and y. Then second order partial differential equation

Rr+Ss+Tt+ f(x,y,z,p,q)=0

Where

r, s and t are

and R, S and T are continuous functions of x and y in domain of xy-plane.

Then if

S^{2}-4RT > 0

Then equation is hyperbolic

S^{2}-4RT= 0

Then equation is parabolic

S^{2}-4RT= 0

Then equation is elliptic

Example-

3 u_{xx} + 6 u_{xy} + 2 u_{yy} = 0

R=3, S= 6 and T=2

And S^{2}-4RT= 36 – 24= 12 >0

Then the equation is hyperbolic.