Choose the correct answer for the following partial differential equation: ∂²u/∂t² + 4∂²u/∂x∂t + 4∂²u/∂x² = 0. Order = ?, Degree = ?, Type = ?

Steps to Solve

1. Identify the Order of the Equation
   The order of a partial differential equation (PDE) is determined by the highest derivative present. Examine the given PDE and note the highest order derivative.

2. Determine the Degree of the Equation
   The degree of a PDE refers to the power of the highest order derivative in the equation. If the highest derivative is raised to a power, take that into consideration while determining the degree.

3. Classify the Type of Equation
   To classify the type of a PDE, look at the coefficients of the highest order derivatives.
   For a general second-order PDE in two variables, $a\frac{\partial^2 u}{\partial x^2} + b\frac{\partial^2 u}{\partial x \partial y} + c\frac{\partial^2 u}{\partial y^2} = 0$:

• If the discriminant $D = b^2 - 4ac > 0$, the PDE is hyperbolic.
• If $D = 0$, it is parabolic.
• If $D < 0$, it is elliptic.
  Identify $a$, $b$, and $c$ from the equation to calculate $D$.

4. Summarize Findings
   After identifying the order, degree, and type, summarize these findings clearly.