Classification of Second Order Partial Differential Equations

classification of second order partial differential equations

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

classification of second order partial differential equations

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


Then if
S2-4RT > 0
Then equation is hyperbolic


S2-4RT= 0
Then equation is parabolic


S2-4RT= 0
Then equation is elliptic


Example-

3 uxx + 6 uxy + 2 uyy = 0

R=3, S= 6 and T=2
And S2-4RT= 36 – 24= 12 >0
Then the equation is hyperbolic.

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