16. Partial Differential Equations – Basic Concepts

Try these first

1. 1.

   Define a partial differential equation (PDE).

   Hint

   Consider how it differs from ordinary differential equations.

2. 2.

   What is the order of the equation ∂²u/∂x² + ∂u/∂y = 0?

   Hint

   Look for the highest derivative in the equation.

3. 3.

   What does the order of a PDE refer to?

   • The number of variables involved
   • The highest derivative present
   • The length of the equation

   Hint

   Focus on the derivatives when answering.

4. 4.

   True or False: Linear PDEs contain nonlinear terms.

   • True
   • False

   Hint

   Recall the definitions of linear and nonlinear.

5. 5.

   Given the relation z = ax² + bxy + cy², derive the corresponding PDE.

   Hint

   Focus on differentiating each component step-wise.

6. 6.

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

   Hint

   Use the discriminant formula carefully to classify.