2.5. Examples for Practice

Try these first

1. 1.

   Classify the PDE ∂²u/∂x² + ∂²u/∂y² = 0.

   Hint

   Check the coefficients and calculate the discriminant.

2. 2.

   What type of PDE is ∂u/∂t - ∂²u/∂x² = 0?

   Hint

   Identify the coefficients and find the discriminant.

3. 3.

   What is the classification type for a PDE with discriminant Δ < 0?

   • Elliptic
   • Parabolic
   • Hyperbolic

   Hint

   Think about the conditions of steady-state systems.

4. 4.

   True or False: A parabolic PDE has two distinct characteristic lines.

   • True
   • False

   Hint

   Consider what a parabolic shape looks like.

5. 5.

   For the PDE ∂²u/∂t² - 9∂²u/∂x² = 0, classify, interpret its physical significance, and solve for the general solution.

   Hint

   Focus on separating variables and considering initial conditions.

6. 6.

   Classify and analyze the PDE ∂²u/∂x² + 4∂²u/∂y² = 0, ensuring you work out boundary conditions.

   Hint

   Think about constant solutions in a bounded domain.