Question

For each of the following equations, determine whether the

equation is elliptic, hyperbolic or parabolic.
(a) uxx − 4uxy + uyy + 2uy + 4u = 0.
(b) 9uxx + 6uxy + uyy + ux = 0.

Answer 1

In equation (a), it is elliptic. In equation (b), it is hyperbolic.

Step-by-step explanation:

(a) The given equation can be rearranged as:

uxx - 4uxy + uyy + 2uy + 4u = 0

This is a second-order homogeneous linear partial differential equation. We can classify it by examining the coefficients of the second-order derivatives. In this case, the discriminant is 16 - 4(1)(1) = 12. Since the discriminant is positive, the equation is elliptic.

(b) The given equation can be rearranged as:

9uxx + 6uxy + uyy + ux = 0

Similarly, we can classify this equation by examining the coefficients. The discriminant is 36 - 36(9) = -252. Since the discriminant is negative, the equation is hyperbolic.