Question

Classify the following partial differential equation as either elliptic, hyperbolic, or parabolic, and reduce it to the normal form: 2∂²u/∂x² -2∂²u/∂t²-4∂²u/∂x∂t=4u=0

Answer 1

The partial differential equation 2∂²u/∂x² - 2∂²u/∂t² - 4∂²u/∂x∂t = 4u = 0 is hyperbolic based on the discriminant method. Reducing it to its normal form requires variable transformations.

Step-by-step explanation:

The given partial differential equation (PDE) is 2∂²u/∂x² - 2∂²u/∂t² - 4∂²u/∂x∂t = 4u = 0. To classify this PDE, we look at its second-order partial derivatives and compare them to the general second-order PDE A∂²u/∂x² + 2B∂²u/∂x∂t + C∂²u/∂t² = 0. The equation can be classified based on the discriminant B² - AC.

In this case, A=2, B=-2, and C=-2, so the discriminant is (-2)² - (2)(-2) = 4 + 4 = 8, which is greater than zero, so the PDE is hyperbolic. To reduce it to its normal form, variable transformations are used, but the specifics of this process are beyond the scope of this answer.