2.4. Canonical Forms

Try these first

1. 1.

   What is the discriminant formula for classifying PDEs?

   Hint

   Think of the standard form of a quadratic equation.

2. 2.

   Identify the type of PDE if Δ < 0.

   Hint

   This type indicates steady-state behavior.

3. 3.

   What is the value of the discriminant Δ for elliptic PDEs?

   • Δ < 0
   • Δ = 0
   • Δ > 0

   Hint

   Think about the conditions for each type.

4. 4.

   True or False: Hyperbolic PDEs have one real repeated characteristic line.

   • True
   • False

   Hint

   Recall the behavior of waves and their propagation.

5. 5.

   Classify the PDE: ∂²u/∂x² + 4∂²u/∂y² = 0. What conditions do you derive from this?

   Hint

   Calculate the discriminant first!

6. 6.

   Transform the given PDE ∂²u/∂x² - 4∂²u/∂t² = 0 into the appropriate canonical form.

   Hint

   Think of the methods required to change coordinates.